If you'd be going round about the 12th then avoid York. The 12th and 13th are race days so everywhere will be fully booked and those that do have availability the majority charge a surcharge for race days.
If you're not planning on those dates the only hotel I can think of with a pool that's walking distance from the city centre is
The Knavesmire I've never been there but a friend of mine has stayed there and loved it. There are probably more with a pool but that's the only one I know of.
There is a nice Spa hotel towards Leeds way but if you want shops then it means driving to them as there's nothing near the hotel.
this is it They do have a hotel in the lakes though which might be nearer shops