Plumbing – Can only use 1 water source at a time? Shower stops running when faucet is on, vice versa

Adjusting the whole-house water pressure is unlikely to solve this problem, simply because you set your shower temperature based on some initial balance of hot and cold. When other people draw cold water only, the pressure difference will still have the potential to affect your shower temperature.

Instead, you should get a pressure balanced valve or shower fixture. These fixtures automatically reduce hot water pressure when cold pressure falls, so instead of a temperature spike you simply get a momentary decrease in output pressure. You can find pressure-balanced fixtures (which would replace your existing shower fixtures) or an independent pressure balancing valve (which you'd want to install near the shower but would allow you to keep your existing fixture).

Any single handle/single spout faucet in effect connects the cold and hot water lines. That's part of your answer. You'll have to do more experimenting with the tub controls and both hot and cold water supplies.

Low pressure itself is not the cause, but low pressure could lead to flow reversal when a tap is open.