Draw offers

I could be mistaken, but I believe in some of my early tournament games (Online chess version) I have offered a draw and it wasn't accepted by my opponent at the time that I offered, but a few moves later, the game had ended in a draw.

Question: once a draw is offered, is that draw offer available for the rest of the game? If I offer one time, and they decline, then a few moves later THEY offer a draw, is it automatically drawn.

Reason I ask is because I have a game against someone where I offered a draw earlier, they rejected and have since made a bad move where I now have the upperhand and I do NOT want the game to end in a draw.

Thanks!

No the draw offer is only one move falid.

But there are other ways to draw, get the same position three times and it is a draw.