No 225 is wider (225 is width in mm). You may want to try a 205/50R17 for winter.
Tire Rack has the Blizzak WS-60s in your size for $133 per tire, within your budget.
Continental Extreme Winter Contacts are $114, same size.
General Altimax Arctics are $91 each. Probably worth paying more for the Blizzaks, but I think any of these would be a significant improvement over an all season tire.