...Could I have won, here? Or was it better for me to have accepted my opponent's draw offer [which I did]?

you have a little vantage but is too small for win better draw and play another game

The advantages I see for you are:

• White's h-pawn is doomed.
• You have a passed pawn.
• Your pawns are 2&1, whereas his are all isolated.

The disadvantages I see for you are:

• White's King is much better placed.
• Your c-pawn is backward and weak.
• White's pawns are more advanced.
• White can force a trade of his b-pawn for your c-pawn.

His current threat is b6, forcing a pawn trade.  There's nothing you can do to stop this, just like he can't hang onto his h-pawn.  That leaves you up a passed pawn.  As my friend Milt once said, "Being up a passed pawn in an otherwise even position is like being up a pistol in a knife fight."

So, why accept the draw?  Unless you blunder, you are not going to lose.  The draw will still be there in a few moves.  Subtle slip-ups on his part could give you a win.  I'd keep going until things are more resolved.

it is true that as long as you play fairly well you will have a passed pawn that your opponent will have to deal with.Also if you worked your rook and king together you could make his king retreat to either the back right corner or back left corner.Either way i would have played a few more moves looking for better position if not for a rook king check i would try to promote my g pawn.

This isn't a draw. You have a win

I agree, this is a winning position.

Thanks guys, I won't make this mistake again next time =)

I'd agree.  If you played well enough to have this position in the first place, I don't think you'd have blundered badly enough to lose.  You have connected pawns in the center, one of which (the passed one) is protected, and the other is preventing his from being passed.  He has only one passed pawn, and you're about to take it.

This is a easy win position. The advantages you have are

• White's pawn is already dead.

• The others are immobilized.

Unluckily, your c pawn is causing a disadvantage of position. The best move here is to capture, just like you said, the h pawn.