You should only wear running shoes for running. There is a flexibility in runners that are not in cross-trainers. You should be fitted for running shoes as it is the most important thing you can wear while running. With snow being an issue right now trail runners would probably be a good idea as they have a grip to the soles, but if you don't want to invest in those then get road running shoes when the snow goes away, go ahead and buy road shoes.
Switching shoes is not a bad thing..I run with 2 different kinds of shoes, both running shoes but 2 different styles..it gives both the shoe and my feet a rest and a little bit of a change...even if I have 2 pair of the exact same running shoe I change them out on my runs.
Good Luck to you..and if you have a running store nearby stop in and have your feet checked out. Most of them employ only experienced runners and everyone's feet require different type shoes.