Should I add an escape register for added ventilation in A/C duct to reduce condensation

The best way to combat the condensation problem is to insulate the riser duct, aka metal duct, that is causing the problem. There are several ways to do the job. The best way is to use a vapor-barrier & spray-on foam insulation, but that can be costly and messy, unless done by a professional. Another is to use Rock-wool to insulate the duct, but often there is not enough room in the wall to use it. The insulation that I prefer to use is a Duct-wrap. It looks like bubble-wrap with small "bubbles" sandwiched between 2 layers of foil-lined plastic. It all depends on how much money you want to spend and how much space there is between the duct and the drywall.
The short-term solution, is to install a duct-wrap. It is available at home improvement stores in the building products department in the insulation section. It generally comes in various lengths at a 4 foot width & can be cut to size with scissors (or a razor-knife, if you are careful). To install it, simply measure the distance between the studs and add at least 12 inches to tuck the insulation behind the duct OR add 4 inches & staple it to the studs like traditional fiber-glass insulation. Cut the duct-wrap to size. You will also need to get metal (duct) tape to seal the seams between the upper piece and the lower piece of duct-wrap. It has a paper-backer to keep the glued side from sticking to itself. You will need to peel the paper-backer off when you install it. The Long-term solution would be up to you. You could hire someone to take apart your existing ducts and insulate them or you could do the job yourself, either way that is the only solution to your problem. And by the way, cutting a hole on the wall will only make matters worse by giving the humidity a direct path to the now un-insulated ducts.

Sounds legit. As you decrease the temperature of air inside the chimney you decrease the flow rate out of it causing the mixing with cooler air to happen in the chimeny rather than out of it. It could be that this didn't happen in 2011 because of warmer outside air temperature, different humidity conditions, or that your exhaust was running hotter/faster then. If you use the formula on wikipedia you can calculate your desired chimney cross section (just remember to use metric to stay consistent). Also the damaged cap definitely could slow your flowrate which could contribute to this issue.