The great majority of houses in the USA have substandard duct systems. The duct system to the second floors are often much much worse. When you put in a zone system the unit has to be sized to be able to run both at the same time, and have a dump zone when only one zone is calling.
The system currently is running three floors with two of the rooms on the first floor with their ducts shut off at the registers.
The problem is often that the system requirements for both zones together is more than what the entire duct system can handle. So, when you cut off some of the flow and capacity your problems become worse, especially to the second floor, where overall static pressure of the poor ducts overcome any flow trying to get there.
That statement would be true IF the duct work is/was undersized a lot. A bypass damper would help in this situation although not optimal.
Also, unless you have returns in each room upstairs, dont shut the doors. No flow, no cooling/heating.
All rooms with the exception of the bathrooms and basement (currently) have returns.
The basement is an entire different biome to the rest of the house. It will need either a mini split heat pump or some other source of heating as well as a standalone dehumidifier. A whole house dehumidifier works very well. Ultra Aire makes very good ones.
Dont duct it with the rest of the house. It will NEVER be comfortable down there. Either too humid or too cold.
I must have gotten lucky with my last two houses because the current one keeps the basement at 68F/51%RH. The old one was a little colder but 47~49%RH but that one had a return. As far as basements go it is how they are built and sealed. The old house I remolded the basement which it never showed signs of water leaks. I spray foamed it and made sure I sealed all the exterior walls. The new house was already finished and has shown no signs of water, plus it was built in 1986 and has an energy certificate with it from the builder (if that really means anything).
When it gets cooler and I can get in the attic I'll know more what I can do to help it up there.
As far as the basement goes, the finish job wasn't that great and the supply ducts are in the ceiling not the exterior walls so I may see about moving them and possibly adding in an extra duct and return that I can open/close.