Uncle Buck nailed it.Â
But it sounds like you've not engaged a lawyer, or you would have already known about apostilles. Sounds like you've figured that out now.Â
I think you got your move actions WAY too far ahead of your documentation.
Don't count on making that mid-April delivery; the government is unlikely to be fast enough to satisfy your needs. Here's some suggestions to help (I hope):
1. I don't think you'll be able to finish your Residency application if you enter the country on a normal travel Visa.  The Residency process is specifically designed to ensure you get pre-screened at the Embassy, and then you get approved for a specific Residency Visa to visit the country to complete your Residency application. THE EMBASSY OR CONSULATE APPROVAL of a Residency Travel Visa DOES NOT NOT NOT afford you ANY ANY ANY residency privileges. If your shipping container arrives prior to approval of your actual RESIDENCY, you'll owe higher duties on the contents. How much? check the other threads on this website.
2. I don't think many lawyers will just write a letter of referral for you if they don't have a chance to review your file and get to know you a bit.  Additionally, YOU shouldn't want to be associated with the type of lawyers that would write such a blind letter -- getting associated with the wrong crowd is not a good thing for new expats to do, in any country.
3. Contact your shipper NOW and ask for estimates for delaying the delivery and storing your shipping container in a port -- stateside or in DR. Hopefully, you can catch it before it departs the US. It shouldn't cost too much for just a couple of months, which is all you'll probably need--but it could be 5-6 months...as others will attest on this forum, Seems kind of random.Â
>>> Or, perhaps if you're buying a house or condo, the developer may have a place it could be stored...but you'll probably still have to pay some rental fees for the containers.Â
>>> Make sure to ask about SECURITY of any potential storage areas, BEFORE YOU COMMIT, especially if your container contains high-value items. I think the reasons are apparent; there are cases where folks' shipping containers get looted en route or in port. PAY THE NECESSARY PRICE TO GET APPROPRIATE SECURITY, AND PROBABLY SOME BONDING AND INSURANCE.Â
4. Planner knows some lawyers that charge reasonable rates for their services. You may want to engage one of them, perhaps for a reduced rate if your residency Visa is already near approval.  I would recommend contacting Planner, get a couple of referrals for lawyers, and start making calls.  Depending on your language skills, be careful whom you engage. If you don't speak Spanish well, you really should pick a lawyer with excellent English skills.Â
>>> Planner probably has some suggestions for whom NOT to engage. I strongly recommend listening to her suggestions.Â
I'm so sorry to be so negative on the above recommendations; but you have gotten the cart before the horse, and it is unlikely to be easy or free to get it corrected.  Move quickly to get things realigned, or your stuff will arrive in port early and you'll have a dickens of time dealing with it.Â
Best of luck, my new friend. Please keep us updated on how your story evolves; my wife and I are still awaiting our Embassy/Consulate approval. Would appreciate your "lessons learned" on this process.
Jim