There is a Thousand Trails Yosemite Lakes Preserve (they DO rent to non-members, or the Zone Pass might be attractive) 7 miles North of the North gate on Hwy 120. They are located on an old lumbermill site on a creek, and have a well stocked store with gas and diesel fuel. It IS 35 miles from there over a pass within the park to the valley floor.
If you don't yet have it, you can buy the Federal seniors Golden Age National Park Pass at any Yosemite gate for $10 lifetime. You need to be age 62 or better. It admits you, and everyone in your vehicle for free. These also can get you 50% camping discounts at National Forest campsites, and Corps of Engineers Campsites.
Now until the end of May (traditional school summer vacation)is my favorite time of year to visit Yosemite. The park is less crowded than in the summer, and the waterfalls from the massive snowmelt this year should be about the best it has been in about 20 years.
On edit: To answer your other question, YES you CAN drive through the park to get to outside campgrounds on the other side, but you will have to pay the entrance fee (or use your Golden Age Pass) If you pay the regular entrance fee, that is good for something like 2 weeks, so you shouldn't have to pay again during your visit. I would not plan on coming East to West from the Mono Lake area on Hwy 120. With the snow the Sierras have had this year I don't expect that pass to be plowed open until nearly July. The only other raod that will probably be snowed in for a while is the Glacier Point overlook access road.
