For the last 3 years we've been going to Florida over Christmas Week (From Indiana). And yes, we have a dog and a cat. Once our son and his 2-year old went with us too.
Here's how we do it. Camper is winterized in Indiana and pack and ready to go. We carry a 6 gallon water jug with fresh water.
We plan a drive about 1/3 way to Florida, which usually puts is in near the Tennessee - Kentucky border. We do a search on the internet for "Pet friendly" motels and they are now everywhere! We also check to make sure they have adequate parking for RV's, and so far, we've never had one say they did not have adequate parking. We do the drive, check into the motel, leave the camper in the parking lot, bring the cat litter into the motel room, and ENJOY THE SWIMMING POOL AND HOT TUB. Because they do not want you to leave pets in the rooms by themselves, when we are at the pool, we put the pets in the camper. That is their home, and actually more comfortable there than in the motel alone.
Second day of travel, we travel, not so far, and have pre-planned an over night at a KOA or a Good Sam campground (It's important to have full hook-ups). It is here we get there a bit early in the day, so we like to restrict the driving to less than 5 hours. It is here I un-winterize the camper using the campground water hook-up and the available sewer hook up to flush plenty of water through the lines to clean out the "pink stuff."
As we usually leave Indiana on Saturday morning, this puts at the motel Saturday night, and our first campground out of the snow belt on Sunday afternoon, evening. Monday morning we leave, having dumped the tanks one final time from all the winterizing and using the camper over night, and adjust our time of departure so we arrive at our destination campground Monday afternoon, early enough we can set up camp. We stay till Saturday morning, and reverse the process coming home.
Coming home, we leave Saturday morning and, usually taking a different route home, we find another campground to spend the night, just slightly South of the snow belt (usually in Northern Tennessee), and on Sunday morning I will re-winterize the camper to head home Sunday and get home Sunday night. (coming home is 2 days).
Now, last year (Christmas 2015), we reversed the trip home completely, Left Florida Saturday morning, drove the same route home, winterized at the same Good Sam campground, stayed at the same motel Sunday night, and made it home Monday evening.
3 years ago, the motel we stayed at had a tornado that night! Now that was real interesting.
2 years ago we unhitched the trailer at the motel and went sight seeing and a movie.
1 year ago, left the trailer and truck in the motel parking lot and was able to walk to a local restaurant and had a really great meal!
All the time we were out and about, the dog and cat stayed in the camper. Once we returned back, they were in the motel room with us. It world out great!