Mary, I would suggest ripping out the grass between crowns, letting it fully dry out and die, and then use it as mulch. Then work on the grass "next to" or too close to the canes. I am converting an old pasture full of rhizomatous grasses and that seems to be the only way to make sure the grass I've removed stays dead. It makes great mulch after that. Sheet mulch away from the raspberry crowns to keep the grass at bay. You might consider, long term, planting plants like comfrey, rhubarb, or other densely rooted plants as a barrier to prevent further grass encroachment. Toby Hemenway has a list of such plants in
Gaia's Garden .
Do remove your old canes that fruited last year to open up your raspberry plants to more light, and better air circulation. If you have had problems with borers, you probably don't want to just cold
compost the old canes but hot compost them ( make sure the pile stays hot for several days) burn them or get them hauled away. Your strawberries may well develop runners into your raspberries but that's a much more pleasant and tasty problem to deal with. I'm seeing my strawberries slowly sending stolons into my Anne raspberries even after about six to eight weeks in the ground last summer. I expect more of the same this year.
Good luck with the grass. Shame we can't eat the stuff!