You need to set your Roaming setting appropriately: SETTINGS>MOBILE DATA>ROAMING > HOME ONLY ... A common mistake by people who haven't had their phones setup properly, is having your phone on AUTOMATIC causes in my experience, an increase in signal issues and dropped calls (75% + more)---also, since your phone is set to roam, its ALWAYS SEARCHING for Roaming towers--a HUGE hit on battery---set to Home only--not saying you'll feel 100%, but you will see HUGE diff in your problems mentioned.
I don't know about this--wouldn't this essentially turn off roaming so you only use your carrier? I have Sprint in Boston, and my phone often makes good use of Verizon's network since Sprint, well, sucks. Further, Verizon added subway network which Sprint does not have.
Would turning it to Home stop it from using Verizon?