I'd highly recommend that you stay in the park itself otherwise you'll be doing a lot of unnecessary driving. The park itself is big. Come in with all the groceries you may need for the week. If you need milk or bread 'usually' the store at Rio Grande Village will have it. The nearest big grocery store is in Alpine approx. 75 miles away - one-way. Come in with full water and empty holding tanks.
Here's a suggestion: If you'll be on I-10 through ElPaso, TX then come into the park via Hwy 90 and 118 which will take you to Study Butte. Then enter Big Bend and head for Cottonwood campground for one or two nights since generators are not allowed. Explore some of that area. It's a very pleasant campground in beautiful surroundings right on the river. The easy hike to Santa Elena canyon can be as short as you want. It's very pretty.
Then head for Panther Junction and the Visitor Center and fuel and continue on to Rio Grande Village RV park with full hookups to recharge your batteries and dump your tanks. You can stay there the remaining time or move next door to the campground with no hookups. The RV park are sites all in a row on pavement. However there's a nice grassy, treed area behind all the RVs for sitting. I've seen worse RV parks. The Rio Grande Village campground is set in a treed area with a more casual layout for sites. It's very pleasant and there are areas for generators and no generators. Close by is a dump station. There's also the small store, a fuel pump (expensive) and laundry and showers. Nice walking areas that aren't strenuous and the scenery is very nice.
Keep in mind that if you're 62+ get the federal Senior Pass and you'll get 1/2 off your camping, not only here but any national parks, national forest campgrounds and Corp of Engineers and free entry into the national parks.
When you leave you can take Hwy 385 out from the Panther Junction area to Hwy 90.
If you want to get back on I-10 east you could go to Fort Davis and the state park there is very, very nice. There's also McDonald Observatory and they give night tours of the skies. Then from Fort Davis take either 17 or 118 to I-10. There's another very nice state park on 17 just south of I-10 called Balmorhea if you didn't want to stay at Davis Mtn. State Park in Fort Davis.
Here's the park's website with lots of good information on the campgrounds, things to do, any road issues and a nice map of the park.
https://www.nps.gov/bibe/index.htmHere are some photos of the campsites at Rio Grande Village and other information:
http://www.campsitephotos.com/park/Big-Bend-National-Park..and reviews on Cottonwood campground:
http://www.rvparkreviews.com/regions/texas/big-bend-national-park/cottonwood-campground-nps-campground