HVAC leaking cold air at unit: seal with tape

It's possible.

And may make sense if you think about how the system is working...

First the temperature near the thermostat goes above the set temperature, which causes the thermostat to call for cool air. The A/C unit kicks on, and starts blowing cool air. Once the temperature near the thermostat reaches the set value, the thermostat tells the A/C unit to shut off. The A/C unit shuts down, and stops blowing cool air.

While the A/C unit was blowing, the 30 year old vent covers were restricting the air flow from the ducts. So while the proper temperature was reached at the thermostat, not enough cool air was supplied to the other areas of the house. So the area near the thermostat may actually reach the proper temperature sooner, because that area had unrestricted air flow and received a bulk of the cool air.

When you removed the vent covers, you allowed more of the cool air to enter the other areas (away from the thermostat). So it took longer for the area near the thermostat to reach the desired temperature. Now, because the majority of the cool air was supplied to the other areas (away from the thermostat), those areas were cooled more than the area near the thermostat.

In a perfectly balanced system, all areas would reach the same temperature at the same time. Alas, we don't live in a perfect world. So you end up with a hot office and a comfortable temperature in the rest of the house, or a frozen office and a comfortable temperature in the rest of the house.

To fix the problem you could try to balance the system yourself, or you could call in an HVAC company to balance it for you (which will likely include installing dampers and new vent covers).

The Tee was left open as a means of a clean out.

Start by using a shop-vac on that Tee and hold your finger over the other end, in the pump (you may have to get the pump out of the way).

Next, snake it out, going into the machine: (e.g., the brush that comes with an EZ-Trap; johnstonesupply.com)

Followed with another vacuuming. Ideally the first go sucks it out; you don't want to just push the clog inside. If it still leaks after all that, you may have a cracked pan.