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Nutsedge control with herbicides is not effective.
The article below will help you.

https://www.gardeningknowhow.com/plant-problems/weeds/controlling-sedge-weeds.htm

Bring your container-grown papyrus plant indoors for the winter if temperatures dip below 40 degrees Fahrenheit. If the papyrus gets full sun, it continues to grow through winter. The ideal indoor temperature ranges between 40 and 50 degrees Fahrenheit. A garage, shed or unheated room may provide the necessary temperature.

Supply artificial light if you can't give an indoor papyrus plant enough sun during the winter. Without enough light, papyrus stalks flop over.

Unfortunately, the only recommended method for controlling this grass is what you are already doing.

The only other thing I would suggest is that you lay a black tarp over the area for a few months. Then when everything is dead under, you can uncover, and re-seed with the grass of your choice.

Here is an article with more information: https://www.gardeningknowhow.com/plant-problems/weeds/controlling-sedge-weeds.htm

Sure! You have several choices for your area.

Here are some articles that will offer suggestions, both annual and perennial:

https://www.gardeningknowhow.com/special/shade/shade-tolerant-flowers.htm

https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-6/zone-6-flowers.htm

https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-6/plants-for-zone-6.htm