Thanks all.
I'm attaching a pic now. If you need me to focus on any aspects of the set-up, please let me know. I really appreciate the guidance.
Now that it's mentioned, I think the faucet cartridge is what's probably the issue. Faucet (single handle) is old needs replacing (cheap partly plastic model that is cracking, etc.). So, I am planning to replace either way. Maybe that resolves the issue once I hook up the new one.
Turning to the pic, the top two valves have flexible pipes traveling up to the faucet. One has some red on it apparently indicating hot, and the other blue. I was wrong above. I turned off the red and the faucet worked when on and all the way to one side but did not when all the way to the other side. Tried to alternate to shut-off the blue, but I have a faulty valve there (water starts fizzing out as I close it, but is fine when open). In any event, not sure if any of this helps in sorting out what's going on.
For your info, lower left valve connects to a dishwasher; lower right valve connects to a hot water dispenser.
Happy to supply other information, but I'm thinking the new faucet will solve the issue.