It won't make any difference. Dreamhost hosts mail differently than web services. Doesn't matter which hosting service you have, mail is the same.
If you don't mind spending more, which apparently if you're willing to move to VPS is not an issue, shed dreamhost mail and point your mx records elsewhere. Personally I use Microsoft Exchange. It's a great value for $4.00/month/user. I pay for 2 users, have multiple domains pointed there and have created many aliases and even an extra shared mailbox that box users can access. It's gotten hard to find the $4 plan tho which is called "Exchange Online Plan 1". I drilled down on microsoft's office site to find a good place for you to start.
Dreamhost is still the registrar and webhost for my domains but with a few DNS entries I no longer have to deal with dreamhost mail issues. I switched the email away at the end of July and so far have had no issue whatsoever.