Choices were certainly limited in certain areas but this resulted in a lot more second cousin and other extended family marriages rather than a lot of first cousin marriages (at least in most cases).
I've traced my family tree back to the early 1600s and in all that time I've only come across one first cousin marriage. Although there have been a number of second cousin marriages and also the same families intermarrying over several generations meaning that there are quite a few more distant, but still related, marriages.
There is a term for this, it is called "endogamy" where the same group of families intermarry over generations. This means that the same DNA is kept within the family for a longer period of time than you would expect normally.
For example, my mum had a couple of distant relations that went over to America in the 1600s. To be frank, there were a lot of families intermarrying each other back then (the population in Maryland and Virginia was very small) and you can still see the results of that in the DNA today.
Many Americans who trace their ancestry from early US colonial families have found this when getting their DNA results. I remember somebody saying that they had colonial ancestry from Maryland and Virginia through both parents. While the parents weren't related to each other in any way over the last two hundred years, they had dozens of DNA matches who were related to them through both parents, and quite a few more who had a DNA match to one parent and a paper trail that connects them to the other.
It is not uncommon for colonial Americans, Ashkenazi Jews, French Canadians and US Cajuns to still show the results of this endogamy from centuries earlier. The same thing applies to certain communities from South Asia as well.