Good to see you revisit your thread to update us.
Although the Eagle Mountains are my favorite part of Joshua Tree, Lower and Upper Covington Flats were a favorite of mine until the Internet raised its ugly head. There used to be a reliable spring just west of Upper Covington Flat, downhill. Same restrictions on not camping within a boundary area, which should be on a topo map (I don’t use electronic equipment when hiking other than a headlamp and sometimes a P&S camera).
Someone mentioned sheep making a mess of a spring. Not my experience compared to wild horses and their kin. Some wonderful springs in the Lake Meade NRA that are just about undrinkable unless one is desperate.
If you are confident in your navigation and rock climbing skills, I would highly recommend plan A, weather permitting.
Water has always been a limiting factor for me when hiking in Joshua Tree. In 1985 I bought a Kelty Serac Expedition pack to carry large quantities of water on some of my desert roaming. The pack is something like 115L in volume. My first trip was a circumnavigations of Joshua Tree. I walked from my house to the town of Thousand Palms (10 miles) where I filled 5 gallons of water containers, which was over 40 lb. From there I hiked through Sky Valley and up Berdoo Canyon, which was just wonderful. From there I worked my way to Cottonwood Campground (water stop) then down past Lost Palms Oasis, then north to Buzzard Springs (water). From Buzzard I walked down just below the north entrance of the Park, which was then a National Monument, and was able to get water from a nice homeowner. Walked back to the Calif Riding and Hiking Trail and then to the spring below Upper Covington Flats. Then back to Berdoo Canyon and home. IIRC it was a couple of weeks. More water means slower walking and more difficulty in canyons and such. I still have that pack and use it occasionally.
I still use these 3 packs below for an occasional nostalgic trip. (left to right) Kelty D4 purchased in 1971, Kelty Serac 1973, and Kelty Serac Expedition 1985.

Most of my trips I now use a McHale. I’m 73 so the most water I’m willing to carry at the beginning of a trip is 3 gallons, but most of my desert trips begin with 2 gallons which can carry me through 3 days in cooler weather and a reliable water stop on the third day.
Keep us posted on your plans.