Is it worth it to get a pair of trail-specific running shoes?

I run on trails a fair bit (maybe 50% of my miles) but I never had trail-specific shoes before. Now with all the ice and snow I am wondering if it would help. I don't expect new snowfall any time soon, but there will be slippery areas for a long time in the woods.

One of my reasons for not getting trail runners before, besides the cost, is that I don't know if I want to have my feet in water-resistant material. I fear over-heating.

Any insight?

Thank you!