I don't know about the battery running down, but I should mention the
following: Winnebago has a Dealer Service Bulletin out on furnaces
for certain '97s. The bulletin says 'built 7/31/97 to 10/9/97' and
specifies Winnebago serial numbers 151107213618 and later.
I think mine is actually outside that range and had the same problem,
however. The bulletin number is 7SB004-98-01B.

The symptoms with mine were that the thermostat would cycle way
too rapidly. As a result the flame was on for only about 2 seconds
out of each 20 second (that's right, second, not minute) cycle and
I never got warm. My thermostat has now been replaced by my
local Winnebago dealer and the furnace works like a champ.
I never had any battery problem and the bulletin doesn't mention
that as another symptom, but with a wacky thermostat who knows.