Please allow me to provide some context regarding Republic’s coverage options. Republic provisions cellular coverage with one of two network partners. One partner operates a GSM network and is T-Mobile. Republic’s other partner operates a CDMA network and is Sprint. Cellular coverage is provisioned with one partner or the other not both simultaneously.
CDMA (Sprint network) coverage in your area does have its weak spots. Worse, with CDMA when a phone attempts to hang onto a weak cellular signal; it does so by increasing radio power, which may very well explain both the weak battery and warm phone. It was your mention of this manifesting itself when using the Internet that sent me down this path.
On the other hand, when connected to WiFi, your phone would prefer that to cellular coverage. Therefore, I’d like to confirm the experience is the same when connected to a WiFi source such as your home network and when out, about and away from WiFi?
Overall, GSM (T-Mobile network) coverage looks better in your area. Additionally, GSM technology doesn’t increase radio power output to nearly the same degree CDMA technology does when cellular signal is weak. I’d like to send you some additional information via private message, if you’re amenable to that.