Sounds very similar to what happened to our Rexhall.
"Smell something?" I said ... Got out, looked around, everything looked good. Got going again and, after a few more miles, it happened again. Did this a few more times and then (finally) it became obvious. An inner rear tire had lost pressure and eventually it spun on the rim enough that it rubbed against the Ride-Rite air bag helper springs and wore them down - hence the burning rubber smell.
Lesson learned of course was that a TPMS (now installed) would have prevented the damage.