You had (maybe sub-optimal) working air, so the holes (unless new) are probably not the primary cause. I imagine your evaporator coil (the coil inside the interior air handling unit) has frozen over.
This typically happens when the airflow over it is restricted, but it can happen when the refrigerant has leaked out of the system. Air restriction prevents warm air from heating the coil, which will eventually freeze the humidity out of the air creating an insulating block of ice. Low refrigerant levels super cool the coil creating the same freezing over.
So, replace (or temporarily remove) the dirty air filter. Open all the vents. Turn your AC off for a long enough period of time to melt the ice (typically 4 to 5 hours, but sometimes less). Then turn the unit on. If airflow is restored, you know the symptom of your problem.
If you don't freeze up again, it was likely an airflow problem you just fixed with the new air filter and by opening the vents. Occasionally the interior fan is not turning on, so opening the interior unit is a good idea for visual inspection.
If air handling seems fine and it freezes up again, call a professional to find the leak in your system, have the leak repaired and then get your system recharged with refrigerant. Many pros like to charge the system and run; but, refrigerant doesn't degrade or get "used up". If you're low, there's a leak. Get them to find and fix it now or risk calls and extra strain on your system (which could shorten the life of your system).
Opening the unit involves removing the tape around the panel (it is typically metal duct work tape, so use gloves to avoid cutting your hands). The tape only seals the unit for air tightness. There are screws on the panel. When done, re-tape with metal tape for air tightness.
Keep in mind that the root cause is sometimes electrical. If so, you'll notice the interior fan not is not turning on. There are additional circuit breakers on the air handling unit, as well as the possibility of burnt out control circuitry, corroded / disconnected wiring, etc. It is a good idea to turn the breakers off and on just to assure it's not the cause.
I'd fix the duct work, but to save you money on your AC bill, it's not a likely root cause.
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I talked to my landlord to inquire about access to the actual equipment and it turns out its on the roof and not happening. However, he said he had the same problem with his unit and his Nest and to solve it he just removed the C wire. A big difference between the Ecobee and the Nest is that Nest has a battery and therefore does not require a C wire. I got one, wired it up with the same config as posted, minus C wire, and it was able to call the AC right away. Some commenters mentioned that my C wire might not actually do what it is labeled as, but it did power on the Ecobee. So either I had a bad Ecobee or it isn't able to handle certain funky situations that the Nest can. Maybe the lack of C wire affects the current draw, but hard to say since I never accessed the equipment.