Sounds like your charging system is not working correctly. But is working to some extent.
I would also -
1) Replace your voltage regulator. It regulates how many volts/amps your alternator puts out. You have what looks like a very old one in there now. It’s a best practice to replace the reg when you do a alt. This is most likely your issue.
2) Replace your altenator and vacuum pump belts if you have not yet done so. If you have make sure they are not loose.
3) Clean up your grounds, a bad ground could be causing a voltage drop, increased load would make it worse.
4) Make sure the wire from the alt to the starter solenoid (Fussible link.) and soleniod to the battery’s is in good shape, has good connectors, and no corrosion.
A old and hard belt will squeal when it slips. A old belt may have worn to the groove of the old alternator. (Same with the vacuum pump.). Now perhaps only part of the old belt is making contact with the pulleys and is slipping causing a squeal. A new loose belt can do the same. Also make sure if you have one, your new belt is not too wide, that it runs down in the groove On all your pulleys. One other thought is if your vacuum pump is starting to go bad (Bearings are hard to turn.) it might be causing the alternator belt to slip and squeal.
When you replace the regulator clean all the contacts and the regulator ground. A new regulator is not going to fix corroded connectors or wiring.
Also never assume a new part works, your new/rebuilt alternator could be bad out of the box. But more likely your old reg or your wiring is not up to *****.
I would start with a new regulator and go from there. Let us know how that works out.