Etiquette Question

If I'm clearly winning a game (it very rarely happens!) and for some reason have to abandon, should I offer a draw or just resign? If it's even I'd resign, but it's been annoying on rare occasions when an opponent who is obviously thrashing me has suddenly resigned.

This is more of a question of "chess morals" so there will be many different answers. I personally offer a draw, but if it is not accepted it really isn't a big deal i can just resign. The only exception is in tournaments, like OTB or any online money tournament. doing it in a tournament may affect the standings and can unfairly change the results.

You can offer a draw.  I have had it accepted maybe 1 in 5 times.  

Don't tell the opponent you have to go before you offer the draw. If you are clearly winning and they understand that, they will probably take the draw.

If you tell them first, they will not draw unless they are polite and I would not count on that.

offer a draw first if you're winning:

if your opponent refuses & you have to abandon the game, you should resign (a loss on time is the inevitable result & it's proper etiquette).

There is nothing wrong with either of the approaches.

Is there such a thing as an "Online Money Tourney"?  Who would enter that? The guy with the strongest, fastest computer is obviously gonna "win" such a thing.

Yes, they exist. Obviously the anti cheating measures have to be strict. chess.com's Titled Tuesday is an example. As the name suggests, it's only for titles players.

Resign and move on.  One game isn't going to matter after you play several more games.

I would offer a draw first. If you have time, then chat why you want a draw.  

If your opponent declines, I guess you can resign.  

I never had to completely abandon a game, so I'm not sure what I would do.

Etiquette is only start a game if you can finish it too.