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There are basically 2 possibilities: either the drain is clogged or the condensate pan under the coil has a leak. If you didn't cut the condensate drain line and suck the pan connection and the drain line out with a wet vac then you haven't eliminated that possibility A leaky drain pan is possible but very rare. Let me know if you need to know where to cut the drain line
This copper piping sounds like a refrigeration line but it's usually the larger line where condensation develops which is why it's usually insulated. Since the 2 are run together the rust may be from condensation.
You could see a small amount of water from condensation on the lines (which carry refrigerant not water). Large amounts of water would be from a clogged condensate drain. The only source of water is condensation either from the AC coil or the lines and possibly furnace combustion if you have a condensing gas furnace (plastic vent pipe). Furnace condensation would not show up in or around the AC coil