If the taps are leaking (and from another one of your posts about this) then I would have to guess that the washers have gone bad. There isn't anything special about RV faucet assemblies aside from their cost (cheap) and the material that they are made of (plastic as opposed to plated metal in any of the regular finishes). Quite often, when one of us has a problem with one or just for a little "bling", we will head on down to our local hardware store (local small one or local big box) and purchase a regular home faucet assembly in the proper size (4" or 8") and in a finish that you like and replace the cheaper RV manufacturer's cheap plastic one. The RV manufacturer uses the plastic ones because of their cheapness and they are lighter than their metal standard household siblings.
If you want to repair it, they come apart just like any other faucet assembly and use standard faucet repair parts (washers, seals, O-rings, etc.). As long as they leak, your pump will run longer trying to build up cut-off pressure and it will kick in periodically as the leak drains that pressure down. Your city water will just plain run (drip) and will slowly (you will be surprised, though, how quickly) fill your gray tank requiring more frequent emptying.
Welcome to the FunFinder club and RVing. Great bunch of people here and the FunFinder is a good unit. The company has had growing pains off and on and like any other manufacturer, they turn out the occasional "lemon", but, all in all, they are, IMHO, one of the better manufacturers out there.
My 2 cents, your mileage may vary...
2 Cats; J-Lo and Ragamuffin :R
2014 Thor Tuscany 40RX DP
2011 Ram 2500 Longhorn CTD HO
2011 Keystone Cougar 318SAB (now gone)
2008 FunFinder X 210WBS (Sadly gone)