What would be the best way to move my strawberries?

I planted them with my asparagus - this is year 3 - and they have just over-run the bed. I've tried to mulch around both, but they've just gone so crazy that it's become impossible. And the weeds seem to have figured out that if they grow up right near the plants they can get away with it because I don't pull them out for fear of pulling out my strawberry plants.

Anyway, I want them out of there. Once the asparagus comes up, I just want to be able to use my hoe to weed, put down a thick, straw mulch and be done with it.

So, I know I should have done it last fall, but it's too late for that now - there are already so many weeds in there - I just tried to weed it by hand and it was near impossible.

What would you do? The plants are just greening up and unfurling. I have no idea which are runners and which are the original plants. I do have beds ready to move them too - just not sure where to even start with moving them.

Should I do something now, or wait until runners start going and just move them? Or something else? Would appreciate any advice at all.