My HTC One X seems to be self destructing after Jelly Bean update
If those problems started right away after the update: did the update process include a factory-reset? If not (and nothing else helps), this might be your chance:

Backup all things you can backup (luckily on JB that's not too big an issue, see Full Backup of non-rooted devices -- but I recommend to have all your apps into separate packages, as otherwise a full restore might bring back your problems). Then do a factory-reset.

Before restoring anything, check if your problems are gone. Then restore your apps one-by-one (just to make sure to not bring back the problem).

As for the backup process: check with Helium - App Sync and Backup, this is the easiest way to go. You can mark all your apps, make sure to check that .apk files are included (by default, Helium only backups data), and give it a go. Helium creates separate backup archives per-app, so you can restore them separately. Make a copy of the backup folder to your PC, just in case -- if restoring via Helium should fail for any reason, this still gives you the chance to use adb restore via USB.

Background: It might very well be that some settings got messed up on update. A factory-reset will erase all your apps and data, plus the Dalvik-Cache (optimized app code -- this might be the culprit if it doesn't 100% match the new Android version), so all settings revert to "default". Your Android system ROM (JellyBean) will not be reverted to what you had before, so no worries for that :)