Jim:
Paris is in the middle of the southern part of the state and not quite in the mountains. You'd be in day trip range of Rangeley Maine or the White Mountains in New Hampshire. It's surrounded by big lakes, all of which have a lot of year round and summer home development around them. As you head north and west you get into more remote country. If you want to check out the Rangeley area--big lakes, the nation's best native brook trout rivers (though late July is getting late), moose, and a nice base with restaurants and such in Rangeley, surrounded by the big woods, I can highly reccommend Cupsuptic Campground on the shores of Cupsuptic Lake.
http://www.cupsupticcampground.com/
If you want something more remote and right on the banks of Cupsuptic River (which stays cold so still has brook trout even in late summer, flyfishing only, like most of the rivers in the area), their remote sites are nice, but no services. Outhouse and dip your water from the stream.
Late July is prime time for being on the coast. Depending on how far you want to drive and what balance of culture (with summer crowds) vs. "get away from it" you want, there's 2500 miles of coast line, though you don't need to drive that far to get there. South of Portland is beaches--you can find campgrounds in Wells or Old Orchard, but book early that time of year and expect a crowd.
Portland is a "big city" by our standards with about 100,000 people. Maybe the best collection of restaurants in New England. My sister is a serious foodie who has been all over the world and says the three best meals she ever had were in Portland. Great art museum, lighthouses, parks on the beach not far away. Not sure about camping sites, but no shortage of hotels/motels. If you spend time in Portland, hop on the Casco Bay Lines ferries. Way cheaper than hiring a charter to get a tour of the bay.
North of Portland, Freeport is a shopping mecca with LL Bean and a pile of outlets. A few miles away it's much quieter on the coast. The campground at Wolfe Neck Farm is really nice:
https://www.freeportcamping.com/campsites/ You're still only 30 minutes from Portland here.
A little further east, there is a great mix of beach and rocky shore. Two long peninsulas end at Popham Beach and Reid Beach, both of which are state parks. No camping at the parks, but you should be able to find RV campsites in both areas. This is striper fishing heaven on the Kennebec River in between the peninsulas, and late July is prime time. Night fishing off the beaches can be really good too.
Another 3=4 hours east on the coast and you are on the Maine/New Brunswick border. If you want to cross the border, Campobello Island in New Brunswick has some of the nicest parks I've ever been to, the east coast's largest tides, whale watching, reversing tidal rapids, and a slew of campsites. On the Maine side of the border is Cobscook Bay State Park, where some of the sites sit directly under eagle nests. No services for your RV, but a great spot. On a week night you might have it to yourself. That part of Maine is remote, beautiful, and dirt poor. Don't expect a lot of amenities, but definitely stop for pie at Helen's restaurant in East Machias on your way east. The Canadian side is a little more developed and wealthier.
Great local beer anywhere you go, especially the Portland area.
PM or post here with more on what you are looking for and I will be glad to help.