My screen is badly cracked, is it possible to get a replacement?

Yes.

Make a appt at an Apple Store.

-> https://www.apple.com/retail/geniusbar/

As long as it is an iPhone 5/5C and in the US There is no sign of 5S being repaired yet

if 5S or other model Apple will exchange for an Out of Warranty unit exactly the same as the donor

US prices