I have central a/c (oversized, it came with the house) as well as a dehumidifier that runs 24/7 but my concrete block house here in Florida is rarely below 70% RH in the summer. My tuner complained when he visited in August.
I've replaced the aluminum jalousie windows that didn't close all the way with double-hung vinyl that does.
I have a chimney that does not close, cracks under the door, and a mail slot for my front door as well as three large dogs who are constantly wanting to go in and out of the house.
A damp-chaser would be like putting a band-aid on a gaping wound.
I don't know how to keep my RH below 60% year round, aside from replacing the a/c. That's going to be at least $6K.
If your CAC system is over sized, that is part of your problem. It is cooling the air too quickly so it is not removing the latent heat (moisture) sufficiently. Over sizing a system is just as bad, if not worse, than under sizing when it comes to removing humidity. If you system is older, it will have a 3 speed fan on the air handler. Since you claim it is over sized, make sure it is on the low speed so it removes more latent heat.
On what do you base your conclusion regarding the sizing?
If your stand alone dehumidifier is "running 24/7" and you are still having a problem, perhaps only the fan is running which is making you think it is running. It may have lost it's charge in which case it may run constantly but do nothing. At 70% humidity you should be emptying the drain pain very frequently or, if it is connected to a drain line, condensate should be running out at a fairly steady rate... if not, the DH is not working either properly, or at all.
All the "envelope" items you mention have to be addressed as 70% RH is not only bad for a piano but also for almost everything else in your house. At that moisture level molds can thrive and then you will have a monster problem.
I lived in S Florida for 25 years so I know what you are up against. As a mechanical engineer I designed many commercial systems for the high rise condos on the beach... talk about moisture! If your home is older, (25+ years) you may not have a moisture barrier between the block wall and the inside wall. You may not have a moisture barrier under the slab either.
In any event, you need to successfully address the RH before you have a situation that is both unhealthy for you and for your bank account.