The Tioga is a spectacular drive, but if it is closed at your time of travel there is little [nothing] you can do about it. A drive through Sequoia NP is indeed a viable option and the drive down the Generals highway is also wonderful.

If your budget can meet the extra expense of staying in the parks then I would go for it. Driving in and out of the parks can take up valuable time, and being able to stroll around and witness sunsets and stay in natural surroundings without having to face a drive at the end of the day is very rewarding. You can find more reasonably priced lodgings fairly close by though so you will have to weigh up the pro's and con's. Booking well in advance is advised for staying in the parks.