I'm sure I'm not the first to suggest this particular solution, but what do people think about the following rule changes to help with the draw "problem" that seems to have stricken chess?

1. Abolish checkmate (and other rules regarding the restriction on placing the King in check). In order to win, the King must actually be captured.

2. When a player's King is captured, that player may still make 1 last "dying" move. If the dying move captures the other player's King, it is a draw.

The most signifcant loss to chess would probably be stalemate, as this would require that the stalemated King move itself into check and be captured the next move. However, these rules could also usher in some incredible saves.

For example, imagine you are way behind in material and your opponent puts you in check. Rather than getting out of check, these new rules would allow you to respond by putting the opponent's King in check, forcing them to choose whether to capture your King (followed by you capturing theirs, creating a draw) or respond to your threat first. Can you imagine how exciting those sorts of games could be, with one player's King in constant check for 20 moves while the other player tries to figure out how to secure the win rather than the draw?

What do you think? I understand this would completely change the way chess is played, but I'm curious to see if people think it could work.

So the player who wants to draw, can stop a check with giving a check instead with dealing with the attack.  The player who wants to win, has no such option

Nope I don't think this will lead to less draws.

The player who wants to win can then just move out of check. The player who wants to draw would need to continue to find checks in order to escape, since their King would still be in check.

It may be simply an aesthetic preference, but there's also something pleasing to me about having to earn a draw based on the threats you make to the enemy King, not because you're not allowed to put yourself in danger.

This is the dumbest thing I've ever read.

All I'm hearing out of this is to change checkmate to capturing the king. (Which will delay the end of the game by one move because you need the extra move to actually capture the king.)

I like the second suggestion though. 

I understand the idea. You may have made a fun game/variant on chess.

But as a change to help avoid draws it will do no good on the higher levels.

Let me illustrate

In the given position black normally would have to play Kf8 and lose the right to castle (or lose his Queen), white could be pretty happy and have an advantage.  However black is happy with a draw and plays Bb4+. Now white can have no edge.

Now I look at the position I see that black has the shot Qa5+ with an instant draw (or white will have to play with a piece less)

You see my point?  Draws will be very common at the high level off your new game just because you give the side who wants to draw more resources to do so.