All the above are good answers. If it were me, I'd drive up and camp Friday night at Horseshoe Meadows. If you did not pick up the permit Friday, head down Saturday morning to pick it up (deadline is noon). Get a shower at the hostel if you want. (If you already got your permit Friday, stay at HM at least until noon, for more acclimation time.)
Head up to Whitney Portal, find a spot in the walk-in hiker campsites. If it is full, see if any occupants would mind sharing a spot.
Wake up and start the hike.
While you are hiking, you should drink about half a liter per hour! (If you don't need to "go" every few hours, you aren't drinking enough.) And have a snack at every water stop. With your altitude issues, you probably won't have an appetite above Trail Camp, so plan on snacking a lot heading up to that point. Take two, and maybe three liters of water when you head up beyond the last water at Trail Camp.
Good luck, and HAVE FUN!
