dymo label and Big Sur address book greyed out

I also have a Dymo LabelWriter 450 Turbo and have used Dymo for years. Although mine works fine after upgrading to Big Sur, I have found their software to be highly problematic at times. I can't say that the following things will fix your problem, but I would try all of them. First, double check Dymo's web site to see if they state that your software is compliant with Big Sur. (The version I'm running is v. 8.7.4.23) If not, ask them when a compliant release is scheduled. Next, do a restart in the Safe Mode by pressing the Shift key on restart. This can take several minutes, so be patient. During this process the Mac does various maintenance fixes and cleans it's caches. Then restart normally and test. Sometimes I have to unplug the Dymo printer's power until the LED goes out and then reconnect it. It seems to do some kind of reset. Beyond that, all I can say is to lean on Dymo. Good luck.

Give the Dymo app full disk access in the System/Security & Privacy/Privacy/Full Disk Access preference pane:

I'm glad to hear that.