Showering and leaving the dog(s) in the car in cool weather is not a problem. I have taken my 2 dogs on solo camping weekends and it was really no different than when my DW is along. Having the extra person doesn't help that much because the limitations are still there. Whether alone or not, you won't be able to do some things you could have done without a dog (hiking on some trails where dogs are restricted, going into visitor centers, going to entertainment events, etc.).
We live where it's cool 90+% of the time. When it's hot it's much harder because you can't leave the dog(s) in the car for more than a few minutes. We never leave the dogs "tied to a tree" unless we are there with them. We have a couple of long leashes that we use for that purpose when sitting around camp. Rangers/Hosts don't seem to mind that even when there is a 6' leash restriction.
Some types of RV-ing are better suited to having dogs. Boondocking with a Truck Camper is ideal. RV parks with closely-packed A's and 5ers is not that great. State and national parks are somewhere in-between.
Don't forget that if you are snowbirding you need to have access to vet services. Unexpected things can/do happen and it's expensive when it does.
Having a dog is a MAJOR responsibility, but we couldn't imagine life without them. We probably wouldn't even have an RV if we didn't have dogs. It's like when people get an RV to spend time with their kids for us.