When soil is very dry, it can be difficult to get water to soak in deeply. The dripping idea is good. As you've seen, if you apply water more quickly than it can soak in, it will run sideways.
You can cut the bottoms off of plastic jugs, like a 2-liter bottle, put a hole in the cap, and insert in a hole next to each plant, or centralized spots between plants. Fill them up and it will take time to drip out through the tiny holes, starting already several inches underground much less likely for water to escape sideways.
Putting something extremely juicy, like watermelon rind, at the base of a thirsty plant will release its' moisture over the course of a day or few, depending on how long it takes the heat to 'melt' it. Usually just a day here, but I know SF is cooler. Bury just enough so you don't start a condo complex of fruit flies. The ground needs to 'get' moist before mulch can help keep it moist. If there is no humus - particles that are organic and moisture retentive, moisture can't stay in the soil long.
I would encourage you to keep adding whatever organic matter your 'property' has, especially kitchen scraps. Instead of the work (and wait) of composting first, they can be slightly buried right in the garden. That will also help hold moisture, and help you begin to develop a population of soil-dwelling microbes that really do all of the hard work in gardening. A spot that hasn't had previous mulch doesn't already have a population of decomposing organisms, and mulch is all brown, so it can take a while to decompose without adding some greenery to it.
Browns + greens = humus, compost, fertility, the part of dirt that's not sand, silt, rock, or clay.
Browns = dry leaves, shredded wood, bark, paper, pine needles, generally not moisture retentive.
Greens = kitchen produce scraps, anything trimmed off of plants while green, nitrogen, generally moisture retentive.
It sounds like it's going to take a long time, but it really depends on the amount of organic matter you can add to the surface (including slightly buried,) and what's already going on there regarding microscopic life in the soil, if anything at this point. You can greatly accelerate the process.
