If your caster wheels are in good condition, you may want to try expanding the brass (?) grip rings on the stems with a flat-blade screw driver; assuming that they have compressed over time, which may be why they're falling out.
If they're not "fixable" you could replace just the casters, and not the sockets. Replacement casters can be purchased here: http://www.servicecaster.com/
If you do that, just be sure to get the same diameter & length stem as on the casters you have now. You can get new casters with different size wheels, different composition of wheels, etc. if you want. (I prefer a 3" rubber wheel as opposed the usual 2" hard plastic ones).
If you go with Mesa Trak-Lok casters you might need to locate them in a different spot on the cabinet as they do require a large, single screw hole for mounting the plate(s) to the cabinet. If you go that route, the Trak-Lok plate may cover the existing caster socket holes, so you might not need to fill them.
Regarding padded covers - depending on how secure (well blocked-in) the cabinet is in your trailer, you'll be fine. If it wiggles around a lot and rubs up against some thing rough/sharp, it'll chew a hole through the cover in no time.