Emotionally getting over silly, stupid, mistakes At one point, I reached a Rook and pawn endgame in the Chicago Open. Winning this game would have landed me in the money. I ended up with a rook and connected a- and b- pawns, while my opponent had a rook and a c-pawn. I was able to push my opponent's king back to the eighth rank, but due to time trouble I actually lost both pawns (lol what). Now my opponent had a c-pawn on the 3rd rank, so he needed five more pawn moves to promote.

I lost this endgame. But I'm reading the Secrets of Rook Endings by John Nunn where he discusses this ending in deliberate detail, so I could have definitely drawn. I haven't gone over a single game from the tournament, because the ones that I lost were all due to silly mistakes; all of my games were won (with opponents LITERALLY begging for draws) and somehow I lost over half of them - mainly due to time trouble, but they shouldn't have been lost regardless.

I can review that rook and pawn ending game if I wanted to and see which moves I made a mistake, but I'm simply too embarrassed. How can I get over losing won games?
No serious chess player has managed to avoid losing in a way similar to that which you described. Failing to play at ones usual ability is certainly disturbing when it happens, but it is part of being human; after all, humans can get tired, distracted, or fall into a panic due to time pressure, affecting their ability to play.

Always remember this while you play: since you're human, you can make silly mistakes! If, during a game, you realize that you've made a silly mistake, you need to get your emotions in check no matter what. Just admit to yourself "ok, I made a mistake" and afterwards go on and try to salvage your position with a cool demeanor. DON'T BEAT YOURSELF UP DURING A GAME FOR THE MISTAKES YOU'VE ALREADY MADE! Letting your emotions take over while you're still playing will just end in a disaster.

After the game however, feel free to be as disgusted with yourself as you please, but if you still have games to play, it's better to take a deep breath, tell yourself that you played bad but will bring down the hammer of judgement on yourself after you're finished playing, and go on to play your games as if nothing happened.

When you finally get some time to reflect over what happened in the game(s), you should once again try to be objective, and try to figure out the reasons why you made your mistake(s). Were you distracted? Were you careless, and if so why? Was it because you didn't know what to do in the position? What can you do to prevent the mistake(s) from happening in the future?

In the specific case you brought up, I think you may be able to prevent yourself from losing in a similar manner in the future. Let me ask you this: Would you have lost the position due to time pressure if you had K+Q vs. K instead of K+R+2p vs. K+R+p? Probably not! Why? Because I assume that you know how to win K+Q vs. K like the back of your hand! Here I boldly assume that you weren't down to 5 seconds without increments (severe time scrambles without increments isn't really chess, it's more of a lottery imo), but rather that you had the time on the clock to actually perform your moves without dropping the pieces. If you lose a theoretical ending it's almost always possible to practice it afterwards until you know it as well as K+Q vs. K. That way you can play such endings perfectly, even under rather severe time pressure.

To summarize: everybody makes stupid mistakes in chess, don't beat yourself up about it during actual games, and try to see whether there is an easy way for you to avoid making the same mistakes in the future by being objective in your criticism of yourself.