I think the issue may be within Thunderbird and not something from DreamHost. I have added that add-on and everything on my DreamHost domain seems to work fine. The reason I think it might be a Thunderbird issue is I have the same password prompt issue.... for one of my Gmail accounts.
You might want to post about this on the Mozilla forums. I have been too lazy to but I think I will take a look around there now that I hear someone else is having the issue too. If I find anything I will post here.
The mismatched domain add-on is still a good idea however. I am not sure it will fix your problem though as like I said with me it is Gmail which would not have the mismatched domain issue.
Here is what I found in a quick search:
For me it might be that I have two Gmail accounts so it is checking twice as often as I think. Do you have more than one email account on Dreamhost servers being checked in Thunderbird or have the update set to a short time?