I looked up the word "cohabit" and saw these definitions:
1.to live together as husband and wife, esp. when not legally married
Webster's New World College Dictionary
(Sociology) (intr) to live together in a conjugal relationship, esp without being married
The first one I think should raise eyebrows, if only momentarily. You could argue "as husband and wife" means "like husband and wife", but I feel that's not the initial meaning one gets when reading that.
The second one contains the word "conjugal", and I'm wondering about this specifically.
The word "conjugal" strictly speaking means having to do with marriage, however I'm aware that there are associations specifically with sexual intercourse. For example the idea of the conjugal visit was, it seems to me, specifically conceived with the intention of allowing a prisoner to have private "intimate" time with their spouse (specifically have sex). So I think this definition more clearly needs to say "sexual relationship" rather than "conjugal" ("sexual" being the word of choice in most other dictionaries for defining "cohabit").
Despite this association with sex, it's very interesting that VERY few of the "official" dictionaries mention anything of this for "conjugal". It's not mentioned by Collins in the same dictionary where it uses "conjugal" to basically mean "sexual". I understand the term "conjugal" may be euphemistic somewhat, however dictionaries don't have a habit of shying away from a word's real meaning.
I guess the important point is that the overwhelming majority of definitions simply define conjugal as "related to marriage or spouses". And so we're basically left to read between the lines about its real meaning.
Is it inadvisable to say something like "a conjugal relationship, especially without being married"? Or is its meaning extremely clear. I guess this kind of goes to the question in the title, "Can you have a conjugal relationship with someone without being married to them?"