The 50 gallon gas water heater is on the first floor. Water pressure coming into the house is roughly 40 psi. On the second floor the kitchen and bathroom faucets are cold when turned on. I mention it, but it might be due to the design of the house. On the third floor in the shower, when turned on it runs cold, then a normal temperature, then it starts progressively getting colder so I have to keep turning it up. After taking a shower, a faucet in the same bathroom runs scalding hot. Giving it a minute, it gets to a more normal temperature. The mixer in the shower is an old design, the faucet is brand new. Water pressure everywhere in the house feels a little weak.
Besides clogged pipes as described by Programmer66, thermostatic shower valves are known for this problem. The cold and hot water side is mixed upstream of the closing valve. In normal faucets, the mixing is downstream of the valves. To avoid cold water to be pushed into the hot water pipes (and vice versa), thermostatic valves have special pressure balancing or one-way valves that could be made with diaphragmas, which are not reliable in the long run being a rubber-like thin material, especially the element in contact with the hot water. A Google search "thermostatic shower valve temperature problem" will present many Infos including DIY videos.
finding water pressure release valve around the water meter, adjust the water pressure to 50 psi,if still same problem, try to change the shower cartridge,or good shower head to the bad one,might be the shower head. Or you can add a mixer valve at the water heater outline to get a constant hot water temperature.
Based on the various change in water temperature with the hot water at the various faucets is most likely due to sediments in the hot water lines and fixtures. My guess is that the sediments are from your hot water heater. The first step is to check the hot water heater for sediments and to remove the sediment if any.
Your hot water heater can be flushed by opening the drain valve and drain about a gallon of the water out of the tank into a bucket. If the tank has a lot of sediment, the water will be brown in color. After several flushes and the water is still brown, you will need to do a deep flush, A deep flush requires draining the tank and flushing the tank (gas off-or vacation mode) as it refills. See video on how to do a deep flushing here.
After checking and flushing your hot water heater, you need to clear the sediment out of the hot water line to all your fixtures. It seems that the shower is the farthest hot water outlet. Open the hot water in the shower and let it run for the amount of time that it would take to get hot.
On all the faucets, remove the aerator/filters and clean the filters, but before inserting it back in, open the hot water and let the water run for a few minutes.
The water flow from all hot and all cold setting should be the same if all stop valves are fully opened. If after performing the flushing of the tank, cleaning the filters, and flushing the line without the filter does not solve the problem, then you still have a blockage in the hot water line.