I do not know what vintage Sunline you have, year/model.
A few things, when the system is filled and the water primed the pump should not cycle a lot. Need to define how many times it did it in the course of an hour. If 2 or 3 and not using any water, this is not great. The pressure is going somehwere
The pump cycling means generally 1 of 3 things.
1.A leak on the pressure side bleeding off the pressure
2.Over a very long time period the air pocket in the hot water heater is dissolving (do not think this is your issue)
3.Bleed back to the fresh tank thru the check valve in the water pump.
Since you said it was wet inside, it sounds like a leak. If your camper has the Pex style fittings, the areas where they are screwed together have rubber washers in them. These joints get loose over time and most times just grab them and tighten them back up will stop the leaks. It is good practice to actually check each of these style fittings every year as over time the do loosen up. Just check them for tightness, it they are loose, tighten, if they are tight great.
I agree with Henry on the tissue paper or use a paper towel at the joints. If it soaks it up, then you have a problem to dig into deeper. If tighten will not stop it, then the rubber washer may be bad
Here is a pic of a pex style fitting Iím talking about.
Here is one on the fresh water incoming line of a T2499
And here are some around the HWH of a T2499. These while screw fittings, have different internals on the by pass kit.
Basically look at every plumbing joint, galley faucet area, bath faucet, water pump, HW heater, outside shower, city water connection, etc
Now to the city water inlet. In that garden hose fitting on the outside of the TT is a check valve. It keeps water from coming out when there is no city water hooked up. It is also a back flow preventer when city water is hooked up. If you push the button looking thing in the middle of it and pressure is in the lines, water will come flying out at you. Or even dribble out if low pressure. Do not push it to just test it. Most have a little rubber O-ring on them that can blow out of position going backwards with flow under pressure, then it will leak big time.
Good luck, Let us know what more we can do to help you trouble shoot this.