On planting the asparagus (elsewhere) - you've not much to lose, assuming you want asparagus! With a little TLC for a year, you'll be no worse off than buying new roots in, provided you don't move any couch grass with them.
On what to plant after them, anything really, provided you had a good go at the couch. You can get that out as it comes up next season. Potatoes may not be the best because you may just let the couch get on with it around them (it will grow through the tubers (or they grow around its roots?) though that doesn't affect their edibility).