Improving A/C performance would either mean relocating ducts or adding fans, or ducts and fans (separate from the central system, probably running continuously, where you want to pick quiet and efficient fans) to redistribute air properly. You mention rooms with high ceilings, but don't mention having any ceiling fans (which will mix the air in those rooms.) The "floor only" ducting is a terrible arrangement for cooling. Hardly a wonder the basement cools better with ceiling-only, but ceiling and floor would improve overall air mixing, and thus balance.
As for the heating season suction,
- if the furnace is fuel-based rather than just a heat pump, it might need a combustion air intake so it's not pulling air from inside the house to burn and exhaust, thus causing the house to suck air in from outside to compensate.
- Likewise any other fuel appliance such as a water heater.
- In addition, exhaust fans in kitchen and bath areas that don't have explicit make-up air (rarely done) will add to depressurizing the house,
- and finally stack effect (warm air rising) in heating season (or poorly-balanced cooling season) will cause exhaust from leaks up high and intake from leaks down low on a poorly sealed house.
A new system on old ducts is limited by the design of the old ducts, if they were designed poorly. If the new system came with new ducts (which would be unusual given the expense, and disruption to the house that causes, but evidently might have been advisable) then taking the issue up with the contractor as being done poorly might make sense.
As a quick example of what you might do with a added ducts and fans (perhaps making use of wall cavity spaces to minimize disruption) some small fans that suck from the basement floor and dump 8 feet up in the upstairs rooms (you don't really need to cool the rest of the air above where people are that much) should help the balance. In heating season you could reverse the flow, though the ideal in heating season would be to draw from all the way up at the ceiling of the high rooms upstairs. Relocating central system supply/return ducts could achieve the same thing, it's more a matter of what you find fits your budget both monetarily and disruption to house-wise.