Signing into iMessage problem

I'm trying to set up iMessage for my grandmother, on her two Mac minis, using her Apple ID. The Macs are running High Sierra and Big Sur.

I cannot to sign into iMessage with her Apple ID. Either I'm told that an unknown error occurred, or the password was incorrect. The password is correct, though--I checked.

All I can think of is that Eset Cyber Security (installed on both computers) could be interfering. What's confusing me is:

The Apple ID is working fine otherwise (Mail is fine.)

I tried to sign in with my own Apple ID. The first time, it worked, so I signed out. I tried again and I was told that the password was incorrect.

I cannot sign into FaceTime with my gran's Apple ID. But when I signed in with my ID, it worked the first time and then it wouldn't let me sign in again.

I have looked at Apple Support's article about iMessage and FaceTime problems, but I am still stuck.

According to Apple System Status, iMessage and FaceTime are online.

Any ideas, please? Thank you.

PS I never had any problem setting up my iMessages on my Mac.