Use your US phone for emergencies only. Do not switch phone on all the time but rather just turn it on once in a while to check for messages.
Get another phone in Europe that is not locked. Get local SIM card per country. Remember to bring passport when purchasing SIM cards.
Change SIM card as often as you travel to different countries. But do remember that most prepaid SIMs have expiry dates if you don't use them within a certain time.