For the heat balance issue remove the cold air return from below the closet (it should be just 2 screws) and inspect the vent pipes coming off the furnace. You may find some kinked, bound up, scrunched, or not attached appropriately. Also inspect under the passenger bed looking for the same (this is where all your water valves are as well).
For the shower, just replace the shower head with one of those
Oxygenics (Amazon) ones.
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