On the contrary...you should never wear old shoes...thye are worn out. If you get a really good pair of sneakers, like Saucony hurricane 7's, they need no break in, there is full support and you are walking on a pocket of air. When shoes are old (sneakers wear out in 6 months during everyday use) there is no longer any support, which in turn hurt your feet, ankles, legs. knees and hips. So invest in a good pair of sneakers, the ones above will cost you over $100 but are the top of the line, and you will not feel a thing.
We wore ours everyday, they make them in men and women's, We did 3 parks 3 times during our 5 day stay. No blisters, no leg cramps, no pain. We do work out everyday ( I wakl a min of 3.5 miles per day) but that has nothing to do with foot comfort...you will be able to go further with great shoes as you will be less fatigued. Keep in mind you are walking on cement...so as much support and air cushion you can put between your foot and the cement the better.
Think of good shoes as an investment. If you are hurting, WDW is not a fun place to be.