Admission to Ph.D. program at U.S. university without guaranteed funding
The way you are asking the question, the answer is a clear no: this is not common at every university, certainly not in every department at every university.  In my department (mathematics, UGA) to the best of my knowledge we have not had -- say for the last decade -- any PhD students without funding (occasionally from an outside source) that is sufficient for them to live on (albeit less than lavishly).  This situation is quite common in mathematics departments in the United States, and I think it is similar in other STEM fields.  In the social sciences and humanities, funding is distinctly more scarce, but certainly many entering PhD students get funding in these areas.

You ask for advice.  To get good advice you should speak to someone who knows more about your situation.  This advice is centered on the situation in mathematics, but nevertheless I think it is worth hearing: I would think at least twice before entering a PhD program that was not committed to supporting me.  The academic job market is very tight right now, but unevenly so: it contracts severely at every career stage, so that the majority of people who had a lot of success getting placed at one stage find themselves having more difficulties with the next.  Not just anybody gets funding for a PhD program: the resources need to be there, and you need to be (in the judgment of the admissions committee) worthy of them.  However, it is not nearly as hard to get student funding as it is to land any kind of post-PhD academic position (tenure-track or otherwise: actually, in many cases it is harder to land a research postdoc than a tenure track teaching job).

If you have a very specific plan -- worked out with at least one faculty member in the program -- for your success in that graduate program, then maybe things can work out well for you even without advanced funding.  Short of that: depleting your own financial resources to fund your graduate career is certainly, on average, a poor investment.  Please think carefully before you do this.