They won't just leave and you can't drive them out. You have to trap (kill) the intruders until there are no more. Then you have to find how they are getting in and plug the holes up. Finally, you have to clean up after the nasty little buggers. That is the hardest part as mice and rats have no bladders so they pee everywhere. If hantavirus is a concern in your area (likely) then you need a a mask and gloves to do the cleaning/disinfecting. Cleanup will require dismantling the trailer to get to places behind the fridge, oven, furnace, under the shower pan, electronics, and inspection for damage from their chewing, and for nests. You will be amazed at where they can get into. It is because of this that fast action is required to minimize the work.
You are right not to use poison as the dead mice can get in hard to reach places and stink. It is a fallicy that the poison will make them thirsty and they will leave looking for water. Most rodent poisons are anti-coagulants. I have used snap traps baited with peanut butter (just a little) with good success in the past. They need to be placed bait towards the wall, along the wall where they run. This year, in the fall I had a new mice problem and used the sticky traps for the first time. They worked very well and I liked the idea that I wasn't using bait to attact more mice to the trailer.
I hate (no, really HATE) dealing with mice and rats but my trailer seems to invite them in (I never store food in the trailer) and they enter where the slide rails enter the underbelly. There is no good way to seal it and have the slide work properly. I have had 2 infestations, once when my trailer was 6 weeks old and I trapped 3 Norway rats and then again last fall when I evicted 5 mice using the sticky traps. When I winterized last fall after eliminating the blighters, I removed everything like bedding, stood the mattress on it's side and removed all the drawers and opened all the cupboards. Of course there was no food anywhere and the trailer was scrubbed. I wanted to make the trailer as inhospitable to rodents as I can giving them no place to nest. I made it through the winter with no further problems.
Over the years I have read many posts about dryer sheets and mothballs but there is always someone who tried them and still had problems. I think those who have had success with them are just lucky. The coyote urine and there is also a product farmers use (can't remember the name (Supercab?, no that's my truck)that seems to keep them away but I don't know if the cure (smell) is worse than the problem.
Happy hunting!