Yes, I would certainly have tried to win that.
Draw or Continue to Fight?

Thanks for helping me out! So assuming the rook trade, then what? Be2 is tempting but ends up trapping the bishop.

You should analyze it yourself and then post your findings for other players to verify or refute. Analyze it out from both sides of the bd and keep a record of whatever you play out.
31...Bh3 seems to rescue your WSB, offhand you're the only one with any winning chances thanks to White's undeveloped rooks.

Actually, there is--or was--a piece win: 31... Bh4+ 32 Kg4 Rg5+. Took me forever to notice it (too busy trying to find mating nets!).

He gets the bishop, but yeah, that would be winning! I never saw that combination. I was a little afraid of my opponent, though.

Actually, there is--or was--a piece win: 31... Bh4+ 32 Kg4 Rg5+. Took me forever to notice it (too busy trying to find mating nets!).
That wins the a1-Rook as well, doesn't it?... after moves like Rxg1, then Kf6, then g5+, then Bh3+ and Rxa1.

Nope:
31. Kg3 Bh4+ 32. Kg4 Rg5+ 33. Kxh4 Rxg1 as you said... and now Black plays Kf6 followed by g5+ (forcing White to reply with Kh5) then Rh1+ (forcing Kg4 in reply) then Bh3+ and Whitre loses his a1-Rook.
Am I missing something?
I guess White could immediately (after Rxg1) play a4 and reply to Kf6 with Ra3... but doesn't that get him mated by g5+ (Kh5) Bh3 followed by Be6 and Rh1 mate?

Check my edit.
31. Kg3 Bh4+ 32. Kg4 Rg5+ 33. Kxh4 Rxg1 34. a4 Kf6 35. Ra3 g5+ 36. Kh5 Bh3 and mate in two by Be6 and Rh1 mate.

Yep, that is mate . Good call! Any comments on the rest of the game?
A few.
1. ... Nc6 is not a good reply to White's first move 1. Nf3... unless you are good at playing the Chigorin defense to the Queen's Gambit, since White can almost force that line by replying 2. d4
Check out this demolishion of a 1963-rated player in fifteen moves:
In your game, the opening transposed to a sort-of Ruy Lopez. I would have answered White's 7. Bxc6 with bxc6 rather than your Bxc6. The doubled Pawn isn't much of a handicap, and it gives you extra influence in the center.
I think you were losing around move 20, but White's 25th move was a real lemon, after which you were now winning.

I think it is worth noting that white's 29th move was the decisive blunder. After 29.Kg3?? you were winning, as AndyClifton and blueemu have pointed out, but 29.Bh4!! would have drawn.
Hey guys, I recently played this game against a higher-rated opponent. We drew. Please analyze the game.
In the end, he accepted my draw offer as I proved that I could force a draw. Should I have drawn him or kept going?