nmgirl, there are a couple of us ladies (maybe more that I'm not aware of) here on the forum considering the very same thing.....traipsing around the country in an RV with our dogs
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I've been RV-ing/camping in some fashion since I was just 17, although obviously not around the country. As jujube mentioned, you can usually get a monthly rate in an RV park much cheaper than staying here and there for a few nights...that is if you wouldn't mind staying put for a month. As far as the type of RV, I have had a trailer and a fifth-wheel and the fifth-wheel was by far easier to tow and maneuver than the trailer. But I never got comfortable towing anything.
I now have a truck camper. If you decide at some point to go ahead and get a small RV and tow your little car, my advice would be to practice a LOT with backing up, and take short trips before heading out into the wild blue yonder. And get to know your RV top to bottom, inside and out, so you don't have to be dependent on someone to help you out when you arrive in a park. Hooking up to park utilities isn't difficult.
Boondocking, dry camping, primitive camping, whatever you want to call it, is something I personally would not do unless there were other campers within earshot. I don't think it is safe for a woman alone to be in the boonies with no one around. One thing to keep in mind is that many Walmart stores will allow RVers to stay a night or two in their parking lot for free.
My finances are limited, but I already know I would want a home base if I decide to hit the road for a few months at a time. I have "toys" that require a garage that I don't want to give up. So there you have my 2 cents worth, nmgirl. I hope it helps a little bit as you ponder a future in an RV. We are the same age, I will be 64 in a few days. Does the "nm" stand for New Mexico? If so, may I ask what city? NM is a state I have been researching to relocate to once I sell my house, probably in the next year or two.