For just a 5 day trip and "maybe" occasional use later, and with a, "$3-5K range," I wouldn't go with an RV. You will have a hard time finding anything that is even serviceable for that price.
Anything that inexpensive will need lots of work (tires, etc) to make it safe and useability is another issue (electrical and plumbing issues). Also, pulling a travel trailer will double your fuel usage
Consider long days of driving and motels for this trip.
Maybe others will give you a different perspective on your plan.
Good luck with your move.