Hi Stacey.
My Dad looked into getting a Verizon phone with SIM when we were back in the states in Aug / Sept. There're only one or two handsets that they offer with SIM cards and they're rather expensive ($100+ and that's with a renewal package).
Also, with Verizon they do offer a call international package (I believe it covers international texts as well), but it reduces the cost of calls to the UK. You can cancel or add the package at any time. However, if you want to use your Verizon phone here in the UK you'll be paying an extortionate amount. I believe it was $1.50+/min as its international roaming.
If you didn't already know Verizon charge 25c to send & 20c to receive international texts, however you can send Verizon customers texts through their website.
In all honesty you're totally better off just buying a UK SIM.
As for handsets themselves as long as its unlocked, but then you may encounter frequency issues (handsets operate on a multitude of frequencies) and I'm not a phone techie, so won't really get into it.
You can get a variety of handsets here in the UK though, but it really depends on what you're looking for in a phone, as to which country would be better to buy from.
I used to be with Verizon and I gave my contract to my Dad when I left and he ended up letting that one run out. When we first went back to the states I just took the old handset back into Verizon and reinstated it (paying $50+ top ups) and again gave it to my dad who let it lapse. Second time we went back to the states we just used my dad's phone (he ended up getting a new contract) and this last time we went back to the states I got a T-Mobile sim card. You pay $100 top up and it keeps the SIM valid for 1 year and in all honesty you'd be hard pressed to use up all that in a 2 week visit unless you were on the phone non-stop or texting non-stop. We intend to keep that SIM and just recharge it every year so that way we have the same number for family & friends to contact us on each time we go back.
Only thing is the T-Mobile sim will not work here in the UK. Oh well.
Saves having to pay a reconnection fee every year and going through that hassle.
Good luck with your decision!