We used to have the issue also, and while we now have the Oxygenic, the same problem of cold blast happened on the OEM one that did not 100% shut off either.
I did solve the problem by installing a positive shut off at the hose fitting on the shower faucet. We boondock a lot and cannot afford to waste water. We do navy showers as it is sometimes called, wet yourself, shut water off, soap up, rinse, shut off, shampoo hair, then final rinse. I had to find a brand that was 100% shut off and did not leak and would fit in the space. I had to unscrew the faucet from the wall, spin on the new valve, put faucet back on then put the hose on.
Short of buying a new RV....LOL. here is the issue with the cold blast and leaking shower. Or at least one explanation on why. I'm assuming your camper has a tank type water heater that uses a one 3 way selector valve on the bottom cold water inlet, and a check valve and T on the top hot water outlet of the water heater as part of the winterizing kit. Yes/no? Assuming the answer is, yes, here is what can happen on that little drip.
The little drip is a small leak in the midst of a pressurized water system. This effect can happen when using city water or onboard tank water and pump. The drip is a very small flow, but yet it is flow and the shower head is open to both the hot and cold water faucet. That drip flow will come from the water pipe with the least effort to allow the drip flow. Now enter the water heater discharge check valve. That check valve has a spring loaded normally closed poppet check in it. That spring may have 2 or 3 lb (etc) of force holding the check valve poppet closed. When the dripping shower head wants to drip water, the water flow it will come from the water pipe with the least resistance to allow a drip of flow. The cold water pipe has no check valve and even if there is only a small difference in pressure drop to get flow, the cold water pipe wins as the flow is smaller then the larger flow which is needed to allow flow to open the hot water check valve when both the hot and cold faucets are open. To get more technical on this, we need to look at all the pressure drop places in the camper. If the cold water pipe had lots more elbows and other restrictions in the water line, then the spring force of the hot water check valve, then hot water line would win and it would drip hot water or a blend of hot and cold.
If you change the winterizing kit to a two, 3 way valve setup, it gets rid of the spring check. That might change the water drip to be a little more warm of a blend as now some hot water can blend with the cold in the amount of the water the size of a drip. How much blend will come from how many elbows and other restrictions are on each hot and cold water line.
Fiddling with getting a positive on/off valve that works at the shower faucet may be the easier fix.
If you want, I can get a pic of my shut off and maybe the brand. It was a store generic brand out of Menards Lumber yard.
Hope this helps
John