Trip to Yosemite

May or early June probably means that Tioga Pass on the west side of the park is still snowed in. Many of the park campgrounds are closed this year meaning that commercial parks will have even more people trying to get in.
Park entry
You should read thru the rest of that park site for more info. And get the partk app on your phone.
Yosemite like Yellowstone and some other major parks is huge, it takes hours to drive places especially if you are camped outside the park. Like many of us Yosemite is our favorite but difficult to get in and far away for us. And worst it is in the evil empire of Cali.
That said my wife won a 7 day pass to a commercial cg on the east so we hope to be there this summer.
I cannot advise you on cg since we always stay inside the park, until this year but you better make reservations ASAP if you hope to find a spot. May could see snow on the ground in the park.

Getting reservations for May or June 2022 will be very difficult. If Tioga Pass is open staying on the eastern side is my choice but the line at the park entrance station backs up early. A number of USFS campgrounds are near Lee Vining but they are without dump stations or power. I like Lower Lee Vining GC. Fresh water is close by.

Just as I was beginning to plan I saw this on a website, published just hours ago

Yosemite National Park may limit visitors this summer due to major construction projects
Multiple road, trail, campground projects could cause gridlock, park officials worry.
In this May 25, 2017 file photo, a class of eighth-grade students and their chaperones sit in a meadow at Yosemite National Park, Calif., below Yosemite Falls. The number of visitors will be limited at the park this summer. (AP Photo/Scott Smith, File)
In this May 25, 2017 file photo, a class of eighth-grade students and their chaperones sit in a meadow at Yosemite National Park, Calif., below Yosemite Falls. The number of visitors will be limited at the park this summer. (AP Photo/Scott Smith, File)
By PAUL ROGERS | progers@bayareanewsgroup.com | Bay Area News Group
PUBLISHED: January 20, 2022 at 3:07 p.m. | UPDATED: January 21, 2022 at 7:09 a.m.
Concerned about nightmarish traffic jams at Yosemite National Park from more than half a dozen major new construction projects, park leaders are drawing up plans that could limit the number of visitors this summer by requiring reservations for day visits.

“This summer is going to be a crazy construction season in Yosemite like you have never seen before,” Yosemite Superintendent Cicely Muldoon told a group of local elected officials and tourism leaders at a meeting late last week. “Bring your hard hats.”

Many of the projects — major road repairs and extensive upgrades to aging campgrounds — have been on the park’s wish list for decades. But funding is available now because Congress passed a landmark law in 2020, the Great American Outdoors Act, providing billions for repairs and upgrades at America’s national parks.

Muldoon told the Yosemite Gateway Partnership that parks planners will know the details of a new reservation system in a few weeks.

s ago.

You have said nothing about visiting Kings Canyon....one of the most beautiful places on earth. It is on the southwest end of Yosemite, worthy of a day+ by itself. The obvious roadside attractions (waterfalls and upward view of Half Dome) are easily managed in a day or too....it's the hiking trails that you need time to explore. Weeks, months....not hours. Climbing the Half Dome trail, the waterfall trails to the top, visiting the Hetch Hetchy and Toulumne Meadows...A couple of days spent seeing the Sequoia National Forest would be a reason to extend your stay on the western side.

The sad thing about Cali is that so many great parks are located in the state. Yosemite, Kings Canyon, Death Valley and Lassen Volcanic to name the top four in my list. I have never been to Kings Canyon mainly because it is hard to get to, you have drive deep into the state and loop around Bakersfield to get to it. We tried one year, were camped outside of it when they closed the roads, maybe a fire.
I've spent months in Yosemite and DVNP and weeks in Lassen. Along with the minor parks like Pinnacles and Joshua Tree.

Adjust your thinking. Approaching Yosemite from the East is difficult, high and windy with big drop offs. The Pass often opens in June, closes by Nov 1.

Your plan makes some sense, to stay outside the Park and commute, especially if you want hookups. Yosemite is loved to death. May is a good time to be there, just approach it from the West.

East is great to explore Tuolumne Meadows. Then cross over and camp on the west to explore the valley. Need more time on the west. No, Tioga pass through Tuolemne is most likely still closed for snow in May. Take CA58 to Bakersfield and head north to come around. To the north you would come over I80 to Sacramento and south on CA99.

https://www.nps.gov/yose/planyourvisit/seasonal.htm

May is a bit late and hot for Death Valley. I plan to go next month while it is still cool.