Are there any plants in your borders? If not, you can simply weedkill the grass by watering on a glyphosphate-based weedkiller, carefully avoiding grass that you want to keep. It will take about a fortnight to kill the grass. After that, you can cover the borders with a weed-suppressing membrane until you are ready to plant.
In that case I think that your only real option is to pay someone to dig it out. Once that's done you could try covering the ground with a weed-proof membrane and covering that in turn with bark chippings - that would prevent any further germination and growth.
That's really hard. I stripped turf off an area to make a bed last year and, even with taking off a good layer and trying to cut deep enough to get the grass roots, I spent last year battling grass. This year is better - I still get grass coming up, but not nearly so badly. I find I just have to keep on top of it or it will establish really quickly.
try looking up no dig gardening on teh web, should gets lots of advice about how to go about suppressing the grass without digging or expense and you should be able to do it without losing the plants you have in by mulching round them.