I find that I get way too nervous about having my f3/f6 knight pinned by the enemy bishop on g4/g5. I hate having my queen threatened like that, and then I hate not being able to move my queen because of the doubled pawns I get upon recapturing the bishop. I think my fear is irrational and causing me to overreact by inevitably pushing the h-pawn, and then sometimes the g-pawn too, or else bringing back an already-developed bishop to intercede. Any tips on how to just chill and not do this?
Well, one thing you may try is moving the pawn to h3/h6 BEFORE the bishop can get to g4/g5. Another thing you can do is alleviate the pin by putting the light squared bishop on e2. This is not a great option as white in the Italian or Spanish game because you'd have to move the bishop a second time to get it there, but it's certainly a good option as black.
Also, if you're in a bad pin, you might want to think about castling long so that doubled f pawns are not the end of the World. In some games, having the 2 bishops is good compensation for doubled f pawns. Also, you can protect the N on f3 with the other N on d2.
But realize this: Your fear is not irrational. A pinned Knight on f3 with a locked out Italian or Spanish Bishop (e.g., with a pawn on d3) and the other Knight on c3 can be a serious weakness that you might want to try to avoid.