From my previous post I said that I decided to root my HTC One and flash the latest CyanogenMod ROM to it. After doing this 4G LTE stopped working (and possibly MMS but I never tested that) but everything else was working fine. After a day or two of Googling around I finally found a thread describing the same issue that I was having.
Basically, the Ting 4G APN data gets erased when you flash the m7spr ROM to the phone. Luckily someone on the Ting forums made a flashable ZIP that you flash after Google Apps, and it actually worked for me with the CM 10.2 nightly ROM. Unfortunately, 10.1.3 didn’t seem to work though.
For CyanogenMod 11 use the lite APN here: http://forum.xda-developers.com/showthread.php?t=2340437
If it does not work you may need to wipe everything and insert the APN in the ROM before flashing. That is what I had to do.