DreamHost made a cock-up, that's known and admitted.
However, one simple step which I would recommend for ALL billing arrangements (not just DreamHost), which I have personally being doing for years - NEVER, EVER, AUTO BILL.
Other companies can (and will) screw up billing systems and those using auto-bill will be hurt by those other companies.
Before you think of jumping ship to another hosting company, please think of the following:
1) Rather large shared hosting companies whose names begin with an B - and the other beginning with a H - Don't offer automatic SSH/Telnet usage - you have to fanny around send off forms of identity - something I personally will never, ever do.
2) Other shared hosts use a cheat (parked domains in a cPanel environment) instead of true unique / individual accounts, meaning that you as the account holder may have to perform many simple administration functions for them - something which can be got round when using DreamHost.
Final note: The recent outburst by the hosting company whose name beings with the letter 'L' is, IMHO, nothing more than a load of old sour grapes.
Don't panic - DreamHost will be here forever! The current mess will be got-over.