Do NOT sleep at the hotel before your hike. I'm susceptible to AMS like you, and Whitney will destroy your head if you head up without acclimating. I had severe AMS the first time I went up. It was so bad I couldn't even enjoy the summit by walking around and taking pictures. I sat there with my head in my hands and wondered if I could make it back down. You will make irrational, dangerous decisions in this state of mind.
If you have two days, spend it at the Portal or even better, the Horseshoe Meadows. The road there is reached from the Portal Road. It's ~10,000 ft. in elevation. There are plenty of spaces for camping there.
You cannot sleep on the ground outside your car at the Portal. Use the money you're saving from staying at the Dow and get an SUV/van rental, or a tent to camp at the Horseshoe Meadows. Cancel both days at the Dow. The Portal Motel right by the Portal Road in Lone Pine charges $5 for a shower if that's what you need.
If you're susceptible to AMS and go up Whitney, you just might kill yourself, no joking. Last thing you want is to stumble around like a drunk with a granite cliff greeting your mistake.
