The red brake light should only go on for a few things. First, if your fluid is low. Second, if your parking brake is on. Anything else? I don't know.
If it is coming on dimly, I wonder if it is just an electrical problem in the indicator circuit.
I would suggest that you try to isolate which part of the circuit is turning on the light dimly. I guess the parking brake sensor must be inside the console? Maybe either unplug the connector or short the wires (depending on how it operates) to simulate the hand-brake being off, and see if the light goes out. Then do the same with the low brake-fluid sensor, and see if the light goes out. If neither of those fixes the problem, maybe you have a chafed wire somewhere. Or maybe something else could be turning it on, but I don't know what.
I think you may have to return that accumulator ball, as I think that the 1993 Thunderbird had moved to the more recent Teves Mark IV ABS system. Easy way to check - if you look at your brake master cylinder, is it attached to a typical brake booster (flat plastic can against the firewall)? Or does it have a black grenade-like object just inboard and below it? If you don't see a grenade-like thing, you don't need the accumulator ball.
Diane at Spinning Wheels is very nice and helpful. I don't think she would try to mislead you. She may just have been confused.