April, what are you actual phone needs while traveling? If you don't need regular access to your Verizon number, then just buy a local SIM card when you get to your destination. Because EU roaming fees have mostly been eliminated, you can probably use the same SIM in several countries for no extra cost. SIM cards are cheap - maybe 20 or 30 euros for enough data to last your entire trip, assuming it's less than a month.
If you really need a US number for people to reach you just occasionally, get a Google Voice number. It's free - you get a US number people can call you on and even leave voicemail. The Google Voice app now includes the ability to receive calls on WiFi or mobile data, plus you can make calls that way too. I have only used their older app, Google Hangouts, overseas, but I assume Google Voice works about the same way in Europe. You can call any phone with the Google Voice app, and calls to US numbers are free, even to landlines. (The other person does not need to have Google Voice.)
There are other VOIP apps like WhatsApp that are more popular in Europe, but the other person has to have the app too.
The last few trips to Europe, I've used my Dutch Vodafone SIM for mobile service. I bought it on eBay (never actually used it in the Netherlands) but have used it in France, Italy, Spain, Portugal, and Slovenia without issue. Last year when I went to Portugal, I added 10 euros to the SIM so I could get 2GB of data, plenty for my 10 day trip. Because I use my Google Voice number as my main phone number, I still got incoming calls on the same number I do at home.
But have you considered alternatives to Verizon for your regular phone service? You might consider Google Fi, which is now supported on many phones. I don't need that level of phone service at home as my phone needs are pretty minimal, but Google Fi sounds like a good idea for many people.