Hi dskogs and welcome to the Lincoln Forums. In addition to the good information provided by Peter: How many miles are on the old tires and how evenly is the tread worn? Once tires are down to about half tread (~5/32"-6/32" of tread depth), they are approaching the end of
usable snow driving.
It may simply be time to replace them in general for on-snow use, as opposed to the quality of the tire itself.
Also...Yes, apples to apples, the wider the tire, the poorer the snow grip. A wider tire can/will begin to act almost like a snow shoe (a loose but apt analogy) and begin to float (EDIT- Figuratively speaking...lol) on snow, as opposed to sinking in and allowing the tread to bite in. And of course, the more worn the tread, the larger the problem. A narrower tire will sink in and bite better...again, apples to apples.
If you live in a heavy snow region and are truly worried, there are options, as Peter mentioned. Snow tires are the best option. And you can save money by buying a set of winter wheels and snow tires in a smaller diameter, i.e. 18" or 19". Or an all-season tire that has better snow ratings.
I would recommend going to The Tire rack website and using their Tire Decision Guide to find a tire that better fits your needs. Or even to get an idea on the costs of a better suited set of winter wheels and tires.
Let us know how you make out and good luck.