Will the Rear end out of a 1988 cougar xr7 fit in a 97 mustang?
Yes, you can use it... and yes, your axles will work.
The 8.8 housings in Fox Body Fords from the mid-80's up to '98 are virtually all the same. This goes for Mustangs, Capri's, T-birds, Cougars, Mark VII's... all except for the cars that got independent rear suspensions.
As I implied, the 7.5 axles are actually the same as the 8.8's in the 94-98 Mustangs, so you will be fine, there.
With regard to the suggestion that you use two same-side Ranger axles, that IS what some people need to use, but it is unnecessary in this case. The people that need to use those are people who are trying to switch their '93 and earlier Mustangs to 5-lug without making the rear track width any wider. In your case, you want to use your stock axles, so that you keep the same width that you had originally... like I said before. The Ranger axles would be shorter, which I doubt you want.
Hope that clarifies the situation some.