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When a plumbing problem is occurring at just one fixture, it's a safe bet to assume that the problem is with that fixture. So even though you have replaced parts, there is still a problem to be solved! From your description, it sounds to me like an anti-scald valve problem. The anti-scald valve is a really cool invention. It (usually) prevents the user of a shower from getting burned (or "scalded") from the hot water when there is a sudden drop in cold water pressure… such as when Junior flushes the toilet in the middle of your shower "aria"!
Some shower faucets have the anti-scald valve built into the mixing valve and others have a separate anti-scald valve. Your best bet to isolate this problem is to take the mixing valve (the part you have replaced) or the instruction sheet to the plumbing supply store and have them pull a schematic of the faucet out of their catalogues. Having the faucet's brand name and/or model number will be even more helpful as some faucets use the same mixing valve for different models! The schematic should show the location and part number for the anti-scald valve.
The first thing to check are all your water shutoffs.. especially if you have had any plumbing work done recently.
The next thing to check for is a clogged aerator. The aerator is the small screwed-on device at the end of the faucet spout that breaks the solid stream of water into hundreds of smaller streams, helping to prevent splashing. The invention of the aerator made small, shallow sinks compatible with high water pressure!
An aerator can become clogged if debris comes through the water lines and sticks inside the aerator. Ever wonder why the water company (if you have a public water system) sends out notices to homeowners prior to line flushing? The purpose of the flushing is to clear accumulations of scale and debris from the large underground pipes. If you were to turn on a faucet, washing machine or flush a toilet while this process was occurring, you could suck some of this debris into your home system and possibly damage these appliances. Of course, grit from an unfiltered well system can also build up inside an aerator.
Fortunately, a blocked aerator is usually an easy problem to repair… just unscrew the aerator and either clean or replace it. Due to the small aerator hole-size in some aerators, the best efforts at cleaning may be futile. Aerators tend to get stuck on the spout due to mineral deposits, making removal by hand a "mission impossible". You might have to use a pair of pliers to get a better grip, but this can severely scratch the finish on the aerator or even crush it! I have tried using various grippers to protect the aerator surface… masking tape or flexible rubber jar lid openers are two favorites… but even so the aerator may still become damaged. If replacement is necessary, take the old aerator to the hardware store and get a replacement. Be sure the thread size on the new aerator matches the old.
Be warned… removal of a severely stuck aerator using pliers can actually damage some types of faucets… especially some poorly designed bathroom faucets with weak internal piping… and force you to replace the entire faucet! There is one last-ditch method to try… use a propane torch to heat the aerator. The heat will cause the aerator to expand and may loosen it. HOWEVER… overheating can cause the chrome of the faucet to permanently discolor. If you decide to use a torch… be careful!
When installing or reinstalling an aerator, always put a generous coat of plumber's grease on the threads. This will make the inevitable re-removal for cleaning much easier!
If the aerator is clear, the only mechanical assembly remaining within your faucet is the "diverter valve". This is a small spring valve located inside the base of the spout. Grit, mineral deposits, chlorine or just old age can cause a diverter valve to stick in the "closed" position, keeping the water diverted to the sprayer all the time!
A two-handled faucet typically has a cover over the top of the spout pivot to gain access to the diverter. You don't have to turn off the water supply to do this repair... just don't turn on the knobs! Some water will drain back from the base of the spout so have a sponge or towel handy.
You can't clean or repair a diverter valve… replace it with a new one.
Depending on your brand of faucet, there may be a schematic of your faucet online at the manufacturer's website. We have links to the larger manufacturers in our Links Library at https://www.naturalhandyman.com/linkslibrary/plumbinglink.html.