Jessica,

Apologies for not replying sooner - I was away for Memorial Day Weekend but neglected to put a hold on getting questions assigned to me.

A good alternative to digging up grasses is to 'smother' them. This means however that you'd have to either move your existing perennials elsewhere and start over once the grass is controlled. I'd be reluctant to recommend them

anyway but herbicides have largely been developed to kill 'broad-leafed weeds' that show up in lawns and these have been designed to not kill grass.

Depending upon the size of the area you are working on you can use newspaper and cardboard, heavy cardboard or tarps. If it's a small area you can use on of the first two options, wetting the paper/cardboard down and then putting soil/compost mix and mulch on top. If you can make this layer deep enough, you could possibly replant the existing perennials back into the bed. For large areas tarps work better - but these need to be weighted down well and you'd not be able to plant into the bed this year. If you take this approach, you could also put in metal or plastic edging down around the bed (a few inches) to keep surrounding grasses from sending runners into the bed.

If you do try to dig up the grass that is there, another option would be to look for 'aggressive' ground covers that might be able to out compete the grass.  I'm attaching a handout on ground covers from the Cornell Univ Extension that may be of help and a link to the Native Plant Trust's plant finder tool:  https://plantfinder.nativeplanttrust.org/Plant-Search