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The two possible causes are 1) actual leak of rainwater from the outside or 2) condensation, which is a normal byproduct of the air conditioning process but should be dripping outside your home, not into your wall! Since the amount of condensation increases as the relative humidity outside increases, you are correct to wonder whether there is a connection between the humidity and the leakage.
You can locate the source of a rain leak by making a visual examination of any weatherstripping or caulking around the outside of the machine itself, or the sleeve that the "works" of the AC slides into. If you find any gaps or openings in the caulk, you can scrape out the old caulk and replace it.
If there is a rain gutter over the AC unit, make sure that it is not blocked. If the gutter were to overflow you could get leakage around the AC or through the air vents in the frame even if the caulking seems OK due to the sheer volume of water!
As I mentioned, the condensate from the AC normally flows to the outside and exits the frame via weep holes in the frame (or pan). Some have preinstalled rubber plugs underneath the pan, designed to allow condensate flow. Some have accessory kits that allow you to install a hose to redirect this condensate in dripping straight downward is not acceptable.
If the condensate exit holes are blocked, water can build up in the bottom of the pan and potentially leak out where it shouldn't! You will have to examine your unit to see where these weep holes are and if they are blocked. The usual culprits are rust or paint chips that drop into the pan and block the weep holes. This could also account for the sudden dripping!
AC units that slide into a preinstalled through-the-wall sleeve are often installed level. The body (and tray) must be installed at a slight downward angle towards the outside. Were the unit to somehow change level, it is possible that the condensate would instead build up in the bottom of the frame and leak into the walls. The only way to know for sure is to put a level on the frame to see if it is level. If shifting has occurred you will need to modify the mounting (either inside or outside) to restore the downward tilt.
To your other issue, the loss of cooling power is definitely attributable at least in part to the increased humidity. Some of the cooling energy is being used up drying up the moist air in your home. Dust on the cooling coils would also cause loss of cooling power, and would worsen with increased humidity as the dust holds the moisture, producing a damp insulating blanket on the coils.
Finally, mechanical problems with the coolant pump and/or a loss of coolant pressure are other possible causes that should be investigated.