All FF will dry out.
If you are lucky, the FF has dried in a way that it doesn't rub inside the magnet gap and voice coil . So you hear it as if it had no FF at all.
The FF increases heat transfer and is damping out the resonance curve. The problem may come from the (passive) crossover, it is designed for a certain impedance, which will change without FF. So the sound may not be as it once was.
If you have someone who can clean magnet gap and voice coil and replaces the FF without charging more than a new tweeter would cost, your speaker may sound different when renovated. Hopefull better.
If your are not used to handle objects within a few thousands of an inch tollerance, better don't try DIYS. This is nothing you will succcessfuly finish the first time you try.
You need some instruments too, generator, test amp, some software and a measuring microphone to document before/ after with a near field response curve, best a distortion measurement. This will tell you how the restoration has affected the tweeter. Otherwise it is no perfect job and the tweeter may sound worse than before.
PS also have all electrolitc capacitors of the x-over replaced. Do not "upgrade" or change foil capacitors, coils and resistors!