I'm supposed to start a narrative on just this topic in another thread, but that missive will take too long to be of immediate benefit to you, so everyone else can wait while I sooth your frazzled nerves.
This not going to be a quick process. You are not going to be able to do all the running around at home, submit your documents, have your lawyer do their thing and move to the DR in the next two months. You are going to see another winter at home...
You only have to gather the documents once, but you are in actuality submitting them twice. Once to the DR Embassy to get your passport visa and once when you get here to start step two. Your passport visa is only valid for 60 days. Within that 60 days, all of you need to travel to the DR to meet your lawyer and apply at Migracion here in Santo Domingo.
The secret is to remain calm, expect delays and frustration and set your sights on short term goals, not the long term result. When you order a birth certificate and it arrives in the mail. WIN! have a glass of wine to celebrate.
The process at home is pretty straight forward and is usually pretty manageable. The process here in the DR is clusterf&@k (excuse me) and will drive you around the bend. Your lawyer will shield you from most of it, but in all likelihood, you will need to travel to Santo Domingo more than once to complete the application here and then just when you think it can't get any worse, it will. At the very end, back to SD you go to collect your DR ID.
If you really want to move here, my advice, do not actually get the visa stamped into your passport until you and your family are ready to move here. All the trips to SD and all the potential delays makes it very expensive to travel repeatedly from home to sign this document or get another picture taken etc. Each hiccup stops the application process dead in its tracks until it is overcome. If Migracion asks for something, nothing further happens to your file until it is provided.
It is important that your lawyer has as much of the 60 days afforded by your passport visa available to them to push your application through. If it were to expire before the application is submitted, guess, what, you get to start over again from the beginning. Some of the documents you need to submit are time sensitive. Your criminal record checks must be dated less than 6 months or you will need new ones.
So to answer your question directly...coordinate with your lawyer. Breath deeply, be patient, this is not going to be cheap, fast or convenient, unless you are the exception and then we'll see you at the bar next week.
There is lots of info available here on DR1 in the legal forum and lots of advice elsewhere. It usually costs more and takes longer than you expect or would like. Remember this phrase, it applies to much more than just immigration here in the DR.
Do not ship the bulk of your personal affects to the DR until your application has been completed and accepted by Migracion.