Yes, get a new electronic stat and it will keep the "swing" or hysteresis much tighter.
The best but most expensive stats are those that are Energy star rated and use a microprocessor. I had to use one of those because our furnace is too big for the trailer and had a terrible problem with overshooting the temps.
An ES rated stat takes into account the cool down period of the furnace after it shuts off. Ours keeps the temp dead on now.
The famous Hunter stat worked ok but let the temp go over by 1~2 degrees.