The easiest way to improve your board vision is by practicing tactics. As a diamond member you have access to unlimited training so take advantage of it. I know the TT can be irritating, but it really does help.
When and how do we beginners stop hanging our pieces?

By taking your time to examine all your opponents checks and captures before you make a move.
As you get better I think it's just easier to follow which squares are being attacked, so where is safe to move your pieces.
I still do it all the time in quick games.
It's because you think about your own plans and forget to consider your opponent's options. For starters, everytime before you move just force yourself to consider every possible capture, every check, every pawn move, every move attacking valuable or underprotected pieces.

The easiest way to improve your board vision is by practicing tactics. As a diamond member you have access to unlimited training so take advantage of it. I know the TT can be irritating, but it really does help.
I'm not sold on that, that helps you recognise your opponents mistakes easily but in my experience doesn't stop you making mistakes of your own.
I think it's a thought process, not just concentrating on the threats you can generate.
Just make sure all your pieces are defended by other pieces or pawns.
For example let's say you play Bc4 in any opening , your next move could then be Bb3. On b3 the bishop is defended properly and cannot be a target for any tactics.
And in fact in a lot of variations the grandmasters play Bb3 after they moved Bc4. Of course when they have something concrete they will make more active moves.

+1 for right before making your move think, after I make this move, does my opponent have any checks, captures, or threats.
Chess mentor and Tactics Trainer are very good for practice
All of the previous posts have good advice.
In addition and most important, and this will take some time. Practice concentrating on the squares that your pawns and pieces control. Eventually your brain will sense where a particular pawn or piece is NOT by looking at it directly but rather sensing where it is by which squares it controls. The same thing holds for tracking your opponent's pawns and pieces.
One more thing, never forget that pieces can move backwards. So when you are considering a move always check for one or more of your opponent's pieces ability to capture backwards on the square in question.

The part of the game concerning that everything is at least defended in a basic way is almost like adding and multiplying numbers in our head as we learned in elementary school, i.e., arithmetic. You just have to really focus on counting how many times your pieces are protected (if your piece can move onto a square of another one of your pieces, the latter piece is protected), versus how many times they are attacked.
What I do is, for my bishops, I shoot out imaginary diagonal lines that it can move to in my head, and see if it intersects with any of my pieces, or squares that I want to move one of my pieces to (same for my opponent's pieces). With rooks, I do the same, but with horizontals and verticals. Knights are tricky, but eventually you should get used to them.
If you can conceptualize that geometry in your head, and remain focused at making sure you "scan the board" for all squares that are controlled, you should start to get good at it. Eventually, it will become automatic.

Yes, everybody does it. If you have a hard time with it, then I suggest you make a ''mind move.'' It works something like this:
Check for general threats before anything else. If one of your pieces is hanging that should certainly tip you.
After a move, the position changes. To avoid you miss a basic threat there, you simply make the move ''with your mind.'' (But not with your hand) When you have made this move, check if everything is indeed quite safe from captures and checks, and quite possibly other counter moves. It already helps if you check a handful of moves ''you would do'' had you been the opponent. If it looks ok after you have actually applied this process, you can go ahead and make the move with your hand.
When you get better, something like this happens automatically. We all still forget it sometimes.
Yes, everybody does it. If you have a hard time with it, then I suggest you make a ''mind move.'' It works something like this:
Check for general threats before anything else. If one of your pieces is hanging that should certainly tip you.
After a move, the position changes. To avoid you miss a basic threat there, you simply make the move ''with your mind.'' (But not with your hand) When you have made this move, check if everything is indeed quite safe from captures and checks, and quite possibly other counter moves. It already helps if you check a handful of moves ''you would do'' had you been the opponent. If it looks ok after you have actually applied this process, you can go ahead and make the move with your hand.
When you get better, something like this happens automatically. We all still forget it sometimes.
SIT ON YOUR HANDS.
If you start seeing ghosts, walk away from the table(OTB) or from the screen online even if your clock is running. Walk around, distract yourself. Go back, most times you will have a new perspective on the position.

Do not sit on your hands. It is bad for the circulation and it makes you look stupid. Just check for threats structurally. It is not the same as just staring at the board.

A very common problem & even some very good players do it !
I notice that you've only played Turn-Based/Online chess on this site ? If so, you're doing one thing right - giving yourself enough time to carefully consider each move.
In one of my earliest games on this site (after an absence from the game of over twenty years), I carelessly threw away a knight for no reason & in another game I blundered straight into a fork & lost another knight.
Luckily it didn't lose either game for me but it made me wake up & go about my moves much more methodically. So now before I hit Submit, I go through a mental checklist making sure all of my pieces are guarded & nothing is left exposed. After a while it becomes instinctive & you don't even know you're doing it so you can get back to having fun.
It must work because I havn't lost a piece due to carelessness since ! Of course I've lost some games but that's another story

I think MY biggest failing lies not in placing a piece where it can be captured, but in trying to execute a plan which removes a key defender. For instance, the other day I was playing a game where the bishop on c1 was defending the knight on g5. As I looked longer and longer at the position, a plan evolved where I could take control of the long diagonal and attack my opponent's castled position. So I moved the bishop to b2.
Goodbye knight!! *%$$#*!!! Always always make sure your pieces are defended before moving defenders away to carry out other tasks.
BTW, I won anyway because the concept was sound, regardless of the dropped knight.

Always look for all possible Checks, Captures and Threats.
+1 Yes this. (including checks, captures and threats available for your opponent!)

I think MY biggest failing lies not in placing a piece where it can be captured, but in trying to execute a plan which removes a key defender. For instance, the other day I was playing a game where the bishop on c1 was defending the knight on g5. As I looked longer and longer at the position, a plan evolved where I could take control of the long diagonal and attack my opponent's castled position. So I moved the bishop to b2.
Goodbye knight!! *%$$#*!!! Always always make sure your pieces are defended before moving defenders away to carry out other tasks.
BTW, I won anyway because the concept was sound, regardless of the dropped knight.
In that case call it a strategic sacrifice instead of a blunder !

MSteen: Another way against this problem: Whenever you make a move, look at the squares that it no longer controls (there is no exception to that: no matter how good a move is, there will always be at least one square that the piece controlled previously that it no longer does). So in the case of your Bb2 idea: make note that it leaves the c1-h6 diagonal, and check if that worsens your position unreasonably.
As as new player I can't believe how often I leave a piece exposed to be taken without even realising I have done that. Does everyone do this all the time when they start and how do you break the habit. I know you should check over the game but sometimes you don't see your own mistakes just like it is easier sometimes for someone else to see errors of grammar or spelling in a written piece of work. I can't see how I can improve while I keep mistreating my men - how do I learn to look after them?