We, too, are a family of three and just made the move from a TT to a class C last spring. We LOVE it. We also were thinking Sprinter when we started thinking C, but as we shopped and researched we realized that a Ford or Chevy chassis was a better fit for us. Both from an initial purchase price standpoint, but also from a repair standpoint. Since we both work full time and are not fulltime RV-ers, each day on the road is precious to us. So we decided that we'd rather have something that could be most likely fixed in just about any town in the US with parts that are in stock or easy to come by. Sprinters are a little more specialty and we felt that while repairs could be made, it would likely involve a tow to further away and/or a wait for parts. Not a big deal if we were fulltiming, but a big deal when the office is waiting for us.
We bought a previous rental, which allowed us to get into our 2008 Winnebago within our budget, and we've been very happy. Even at 31 feet we have been able to get into state parks without a problem. We do need to be more selective about sites than with our TT and of course with our PUP, but we haven't been completely shut out yet.
And just a future warning...you mentioned not hitting RV parks - just make sure that you never expose your little one to one, otherwise you'll hear the whining for game rooms, pools, wifi etc. when you're at one of your beloved state or federal spots (speaking from experience here!) We balance out our trips to take turns pleasing everyone, but will certainly head back to more natural settings after DS ages out of coming with us.