I don't think you have a "big problem". It looks to me as though you didn't read the first support message you received very carefully, or that you just didn't understand it. I paid $20 to keep my account alive for another month and asked if it was all good ... [/quote]Paying the $20.00 had nothing to do with "keeping your account alive" for another month - as the support representative told you:
"You can pay in any increments you like as our system simply stores the positive balance as one big pool of money until your
read to pay. You can wait up to 60 days after your renewal date to pay the full price... "
So, you could have paid nothing, $20.00, $40.00, or even $119.00 (one dollar less than what you would owe fro the next year, assuming your $120 number is correct) and it would make no difference - you have a 60 day grace period from your expiration/renewal date within which to pay entire amount that is due. Whatever you have already paid is credited, so you could pay in pieces if you want, but you don't have to.
That sounds like an "automated 'your account is about to expire' warning" that just makes no mention of the "grace period" DreamHost has already told you exists.
In fact, per the support representative, he has already "extended" that grace period for you all the way up until June 20th:
So I think you are "OK", but if you are worried about it, you should write support again and ask for them to clarify it for you