You're smart to look for ways to improve the drainage here, since the number one cause of a failed retaining wall is poor drainage. In a perfect world, weep holes in a retaining wall should be vertical, about two inches/5 cm in height, spaced six inches/15 cm above ground level, and not less than 12 feet/3 m on center. I'd put at least three in each wall, since they're so short and you're having issues.
You really need a four-inch/10 cm diameter drain tile/drain pipe behind the wall (at the base of it), running the length of the wall (this is what typically feeds the weep holes). Without this, your new weep holes will most likely clog with soil/silt from behind the wall in a year or two; if there's no gravel behind the wall, then they'll also certainly weep soil as well as water, which may be messy.
I can see that the slope has a 1-2 foot gradient, but how tall is the wall? If it's taller than 4 feet/1.5 m, you could be looking at a failing wall sooner rather than later. And even a 1-2 foot gradient can cause problems if it's long enough, and 20 feet is certainly long enough. I've seen 2 foot high walls fail after particularly heavy rains.
Assuming that you don't want to get someone dig up the area behind the wall to install drain tile, you may be able to help things by sloping the hillside away from each of the walls, towards their edges. If the wall is not very high, you could also try a variation of a French drain behind each weep hole - dig a narrow cylinder or square channel down from the topsoil to the base of the wall, directly behind the wall and centered on the weep hole. Line the cylinder with a porous landscape fabric and fill the cylinder with washed (not base) gravel. you must keep the top of the gravel clean (with more landscape fabric) to keep soil out of the cylinder. If you slope the hill towards each cylinder, then the rain should enter the cylinder and exit the wall at the weep hole. This will also keep the hole free of topsoil.
If I were in your shoes, I'd probably add the weep holes and hope that the drainage improves (it really can't get much worse). If the wall fails, then you'll have decisions on if and how you want to replace it.