ouch, it sounds like you should be shopping around for a new registrar.
How much do they charge for making the changes? If it is only a small token fee, then it may very well be advisable to simply point the domain(s) directly at the DreamHost name servers, eliminating ZoneEdit entirely. It is not like you will be changing hosts that often.
That was for a situation where you have a domain registered at DreamHost, but the hosting was elsewhere, your situation is somewhat different to this.
Unfortunately, I am far from an expert in these matters and I am unsure exactly how to configure things correctly in your particular situation. I am reluctant to offer any advice that may lead you to more frustration and/or problems. Hopefully someone with more knowledge on the subject will chime in with some suggestions.
Personally, I would probably take the simple approach and configure the domain(s) to use the DreamHost name servers directly, regardless of the fee, and (if possible) I would also be looking at transferring the domain(s) to a registrar that allowed me to change the name servers myself.
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