If I am playing blitz or correspondence and I run out of time but have mate in one can I declare a win? Same with...mate in 3 if I can prove it?

Are you sure? I was playing in a mini unrated tournament. My coach was giving a younger student advice. I lost a queen then some time after he lost on time but had a mate in one. The student was crushed so apparently he made up that rule.

There is definitely no such rule. You lose on time, you lose on time.

Well, I thought it was true because I could draw on time if the position is a forced draw.

No, you are misinterpreting the Insufficient Material rule related to time that states, if you run out of time but your opponent cannot mate you through ANY legal series of moves, the position is a draw. Even in positions like K+R vs. K+R (which you would consider 'forced draw') when you run out of time, you lose.