I offered a draw because it was 5 a.m. (or thereabouts), and I was going to the Treasury. Even if I didn't find anything there, I would have been running around for a long time... :)
I haven't offered a draw to anyone else when I was in a losing position.

And there's no need to offer it... What kind of practice is this?? :confused: It's understandable when one player is a "pedestrian" (at most with a Town Portal) with a small army and weak stats, and the other has full control and is relatively weak... The first player can't catch the second, and the second deliberately avoids contact that would lead to a loss. It's a stalemate where a draw is inevitable... But offering a draw after running around the Treasury for 3-4 days and not really knowing what each player will encounter, and the game situation can change drastically every 2-3 game days - what's the point???? :confused:
If you want to rest, or you're not thinking clearly, postpone the game... No one forces you to play until you pass out.