Can I avoid the slab and run the new PEX through walls and interstitial space?
Good for you for replacing this crappy old polybutylene pipe system. There's no reason at all why you can't do what you're thinking of. One thing to keep in mind is that since PEX is just plastic, it would easily be penetrated by nails or screws driven through the drywall in the future, so you want to make sure to run the new lines at least 2" away from any interior wall space that people might drive nails or screws into. For any places where you have to run it closer to the surface, you need to nail a steel shield plate into the part of the stud that the pipe goes through.