: The faucette in the bathroom is the type w\only one handle that moves right or left for hot-cold. When ever water is running anywhere else in house, the cold water runs at this location. This bathroom is located in the upstairs of a rancher home with a full basement.
: I am planning on replacing the entire fixture. But I not sure this will solve the problem. Any suggestions.

That does sound like an odd thing. The faucet should work for that sink (lav) only. A new faucet may do it. Terry