Do you possibly have a winterization valve partially open? Or another faucet (like your outside shower) that have the knobs on, but the button pushed to off on the showerhead itself?
My money is here. This is a classic cold water pre-mixing symptom.
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Lou & Freya the wonder dog
2008 GMC Sierra 3000HD Allison Duramax
2019 Flagstaff 8529FL