Generally speaking, AC systems make lousy dehumidifiers. They do dehumidify, but they do so as a condition of cooling your air. Here's how it works. AC systems take warm, moist air, and cool it. Water is released when that happens, which then gets collected in a condensate drain and pumped out of your house.
Your new, more efficient system is going to run less, by design, than your old system. The fact that it's running less means it's not moving as much air, so it's not dehumidifying as much as your old system. A more permanent solution is a whole house dehumidifier. It's a separate piece of HVAC equipment built into the system. Some models can remove as much as 90 pints of water a day.