Installing CM 13 OTA causes so many app crashes that I can't do anything on my phone. Can I save my data?
Try booting into recovery mode and enabling MTP, and then get your data from there.

What happened was that the Gapps were incompatible with your OS, since you were using Lollipop gapps on Marshmallow. Once you flashed Gapps, it was already too late, since they need to be flashed on first boot.

Once you have your data, factory reset and flash the packages in this order: CM -> Gapps -> Whatever else (SuperSU?). Make sure that they are flashed one after each other. Do NOT reboot until Gapps is flashed.