iPhone keeps crashing after upgrading to 9.0.1

[Repost of earlier report, that got truncated]

Since I upgraded my phone to iOS 9.0.1, I've had trouble with apps crashing. (I'm using an iPhone 6, and it's paired with an Apple watch, if that matters.)

The behavior:

• Every once in a while, the phone starts behaving poorly. The keyboard and touch screen respond very slowly--e.g. I'll be typing and the keyboard won't respond, then a bunch of letters will all show up at once (so it's registering the touches, but handling them slowly).
• After a little bit of this, the app I'm in crashes and I go back to the home screen--there's usually half a second or so of black before I see the home screen. (This happens with any kind of app--I've specifically noticed it with the built-in apps Safari, Messages, Music, and Podcasts.)
• Once this crash happens, all apps behave badly. If I try to open an app it's likely to load very slowly and then crash me back to the home screen. Sometimes the phone will be unresponsive for a few seconds, then show me the apple loading screen, then take me to the lock screen.
• Killing an app (by going to the task switcher and swiping it up) does not fix things. Rebooting the phone does help--but the problem comes back after a while.

The phone is basically unusable to me now. For example, I just went on a three hour walk where I was doing nothing with the phone except listening to a podcast and tracking the walk with my Apple watch--and the phone crashed this way four times. If I try to multitask (e.g. listen to audio while switching to another app) the crash is much more likely to occur.

I tried doing a full reset to factory settings and restoring from a backup, and it didn't solve the problem.