LTSteve's summary and advice is really good.
The situation here in the DR changes slowly but sometimes dramatically. There are many expats in country who have not sought formal residency and have been here for a very long time. Each time they leave the country they pay a nominal "tax" for overstaying their tourist entry visa.
If you choose to go this route, yes it is easier and cheaper in the short term but you have no formal status. Without govt ID, driving could become a problem. Procuring services such as cell service, some banking services and most interactions with the govt. can be pain.
For me, the biggest kicker would be that after being here for a length of time, the rules change and you either have to leave with all your stuff in tow or scramble to comply with whatever the new residency requirements are at the time. If you are coming for a year and then plan to move on to somewhere else, I'd be tempted to forgo the formal residency application and just put up with the inconveniences of not having a cedula. If, however, you are committed to living here for the long term, you may as well save yourself some potential grief and seriously consider formalizing your status so that you need not worry about having to move unexpectedly, leaving that perfect island home you spent time finding and uprooting your family.
There are rumours that change is coming for undocumented foreigners living in the country. To date, we do not know when or exactly what will be involved should these rumours bear fruit.
As LTSteve said above, the residency process is initiated in your home country at the the Dominican Embassy. Contact them and they will advise you further.
Good luck and maybe we'll see you on the beach.