Once oil has been used for years it is hard to change to something else. Clean well with Naphtha and Then evaluate how dry the wood is . My guess is that more oil is the way to go.

use what is on there, and any stripped spots may not match due to aging. if it works, don't fix it. pure food grade mineral oil does not age/oxidize/go rancid as many other such oils and waxes. cheapest is at Dollar General, Family Dollar, or whatever your local cheap sundries store is named.

...museums use. Good for protecting a variety of surfaces. I use it on knife blades and scales alike. A jar will last forever. Renaissance Wax is the actual name, but it is commonly abbreviated.