There are a lot of if(s) involved with overstaying and a few additional ones if planning to apply for residency.
You must be in the country legally to apply for residency. The residency visa in your passport gives you some wiggle room to get your paperwork submitted if that ends up taking longer than 30 days. You may not apply for residency without a residency visa in your passport - entry by way of a tourist card is not permitted.
You have to submit copies of every page in your passport as part of your application. DGM uses this to see what your entry/exit history for the DR is like. The somewhat newly upgraded computer based immigration systems at the airports gives the authorities another means to scrutinize applicant comings and goings. It is possible that questions could be asked if there is a pattern of overstaying - it's one of those maybe or maybe not situations. Certainly an issue for your DR lawyer to tackle should it arise.
Those contemplating applying for residency in the DR should begin to take steps early on to ease the application process.
1) If so inclined, stop overstaying as soon as you can so that there are no recent immigration violations to provoke DGM.
2) If you are inclined to partake of herbal medicines specially but not limited to Mary Jane, weed yourself off so you can pass a drug test here. Those who ingest lots of medicine might even consider springing for a private test at home to ensure they get the results they need to obtain here.
3) Keep your nose clean until your application is approved and your residency card is in your hand. This could take up to a year.
4) Keep copies of all documents submitted and all documents/receipts given to you forever. Your lawyer will probably insist on keeping them for their files, but they can have copies, you should retain the originals if possible, copies with legible Govt stamps, if not.