I just rode most of this route (Canadian border to LA) this last 8/30 - 9/28.
I can't speak to your original question about the heat & water because I think you were referring to the last 250 miles that I missed and because I was fortunate enough to encounter a rare late September cold snap when I came thru SoCal, mornings in the high 40's & highs of 65°±. Although I will say (particulerly in SoCal) there were several stores marked on the map that were closed, do not put yourself in a situation where your well being counts on the next place on the map being there.
But I would like to address a couple of other things. You say this is your first major trip, I have no interest in dissuading you from this route but it is a tough one. All the climbing can feel pretty relentless after awhile and the pavement on a lot of the descents is crap which can make them pretty intimidating. Actually, a lot of the pavement is crap period but it's scarier at 40 mph than it is at 7 mph. Which is to say I hope your competence as a cyclist is very high and your fitness level is high.
The other thing I will bring up, since it sounds like you have some flexibility, is to leave in the spring from the south. As I'm sure you're aware California has been going through a record drought for years now. As you go through it the entire state feels like a tinder box. Short of a very wet winter and spring this year it's likely to still be ready to burn next year. And September is prime forrest fire season. This year for much of August the "Hat Creek" fire was burning, when you go down 89 north of Lassen you will be riding along it's path for miles. In late August and early September there was a big fire just south of Yosemite. There was the huge "King" fire right when I was coming through Tahoe in mid September. You get the idea. I have perhaps never been so lucky in my life as to the way I threaded the needle on missing the effects of these fires. I had some smoke in the afternoon in Sierraville north of Truckee and some smoke in Yosemite when I dropped down into the valley. In both cases in the morning it was gone. You should not count on being so fortunate.
If you can leave in the spring from the south, keep an eye on the snow pack in the southern Sierra's over the winter, I would think Ca DOT has a web site listing when the passes are open and I would leave in mid April if you can. The rivers and creeks will be charged, the waterfalls in Yosemite will be flowing and you'll beat (at least some) of the crowds there and by the time you get north it will have warmed up.