I found that as soon as I get Swype installed, it works fine, but completely breaks my network connection.
Mobile Data doesn't work. I called VerizonWireless and they were about to tell me to wipe the phone to get the network back working, but first they suggested safe mode. Mobile data began working again. I went back to normal mode and Mobile data still worked. Then I uninstalled Swype and re-installed (the only way I could get Swype back working - changing the keyboard options didn't do it). Swype keyboard now working again, but mobile data again gone.
Probably some bug in Swype that affects the HTC One Remix.