Suggest start looking for a set of snow tires, mounted and balance on steel rims from craigslist or other outlet, might be slightly used but can be a bargain, I've gotten from $200 to $475 for mine, only got 1 set that was a little less desirable but included the steel rims, of course I asked/got a little discount on that set. Than passively look for something that would match what you need for summer tires, either 2 or 4 depending on what's left in the budget. Nothing beats having real snow tires for winter, and not having the air leaks alloy rims seem to get during the winter.
Tires were so rotten on my new winter beater, I got a set of 2 all season junk yard tires 225/70/16 $65 plus $65 to mount and balance, date code was 2020, I needed them immediately, originals were worn to the steel belts, and then I splurged $245 for 2 Champiro Snow tires from Discount Tire. 2012 Kia Sedona, plan is to rotate them front to back as the seasons change, and it won't get much summer driving, and mostly 15 minutes to/from work for winter. Thing to be aware of is especially when you have wide flat tires is the rear will swing around if you try to stop quick, I don't thing the all season will be as bad as the stock tires I learned this on a Toyota Matrix, and I drive controlled most all the time unless there is no other traffic, then I might push some limits, mostly to burn carbon or test traction, not that the thrill is gone, my vehicles might have to last a long time.
My buddy says to me if you want good tires you gonna pay, I'm just a little more frugal but that's about all I'd want to spend and think I'm doing good enough. Sure I'd like the tires big money can get, but I'm more inclined to make that money resource stretch a little farther.
Disclaimer: Last year I bought 4 new General Altimax Arctics for the 2005 Camry, but I don't like using that for I just can't bear it rusting out before nearing 1 million miles. It's being regulated to back-up winter duties for now, and I've already spent the $50 rebate.