$10,000 total for four months is about what we spend - $2500/month.
Some months, gas is low and campgrounds are high. Other months, it is the opposite. Some months we eat out a lot. Some months we take expensive tours and eat out less. Over two years, we have consistently spent $2500/month. We could do it cheaper, but we choose not to. Our budget is fluid - in May, we spent $1000 on campgrounds but only $600 for fuel for our bus and toad. In May 2015, we spent $1500 on fuel getting to Alaska and $400 on campgrounds. As long as we are at our budget for the month, we don't really care how it is spent.
As for maintenance, we average about $250/month. Maintenance includes everything, not just oil changes and tires. We wash the RV at least once per month ($50) and wash the car at least twice per month ($20). We also change the oil in our generator, buy the various greases, lubes, and oils needed to maintain everything in working order, replace water filters, replace outside chairs, etc. when they break, occasionally a coffee pot (we are on our 4th). Maintenance even includes replacing cell phones or batteries. Honestly, I think maintenance is the most important part of full time RV living. If you keep your rig in good shape you won't be left on the side of the road. And, dragging your house down the road at 60 MPH will cause some wear and tear.
Best of luck and I hope you enjoy your trip! There are beautiful things to see out there.