What is the current HTC phone you use? Many of Verizon's LTE enabled phones support GSM as well. Once unlocked, they can usually be brought over to another carrier and activated against their network.
You were not told incorrectly. Verizon is a bit weird, now that they have LTE. The CDMA 1XRTT and E -DO Rev. A 3G networks are still the core foundation of Verizon's network for voice, data, and SMS service. However now that Verizon is rolling out LTE, a technology based off of GSM, they are moving closer to the GSM model. That's why phones from Verizon now have SIM cards.
Here's a spec list of your phone:HTC One 4G LTE
Notice your phone does support GSM, meaning it will work with GSM carriers. Just note that it will only provide 3G access and not LTE access, since many international carriers use different LTE bands from what is used in North America.