I think your adviser brings up a good point that doing a Master's first is a viable option, but I don't think that means you shouldn't apply to any PhD programs. If you know where you want to do your PhD then why not apply and apply to one or two Master's programs as a plan b. My stats were very similar to yours, I had research experience, I applied to both Phd and Master's programs, and I wound up doing plan b. Was I thrilled? Not really, but it beat sitting around for another year waiting to reapply. Perhaps you will fair better than I did =)
Something else to consider with a Master's degree (and this is based off what I learned while researching Clinical Psych PhDs) is that 1) a Master's in Clinical Psych is mostly useless because it is does not allow you to earn a license to practice. 2) a Master's should be something that you can specialize in for your Clinical training. For example: If you want to specialize in the lgbtqi community, counseling/psychotherapy, children/adolescents, neuropsychology, health psychology, etc then you can target your Master's to one of those groups. Be sure to incorporate research training and teaching experience as well. Depending on the school you end up doing your Master's you may have to seek out these opportunities entirely on your own. 3) if you're going to get a Master's in general or experimental psych and want to customize your program to your own interests be prepared to have to explain to everyone, every single time, exactly what your degree is. There's a huge difference between having a degree that people in the industry instantly understand and one that makes one wonder what exactly you studied.
Does a Master's help? Yes if you do well in the program and obtain new skils and/or improve skills you learned in undergrad. What won't help is if you earn a Master's and your gpa/gre is lower than undergrad and you don't do internships, teaching, or research. Another upside to a Master's first is it can allow you to earn decent money on the side. I know a number of people that went this route and are now in full time PhD programs. Instead of spending a few hours a week working some crappy job for minimum wage they're teaching 1-2 classes at a community college. Some of them even teach a full course load during the summer if they don't have their own classes to do.
Speaking of funding....how will you pay for your Master's you ask? You're right the school generally doesn't offer funding, so your best bet is to find a program that is designed for working adults, enroll full time (I mean at least 12 units per term), and keep your day job. I've done this and while it isn't thrilling to have classes at nights or on weekends, it's only 2 years and I have good time management so I'm doing fine. You can apply for scholarships/grants, but accept the fact that you may have to take out some loans if you want to finish in a timely manner. At the PhD level make sure you're funded so that the end of your borrowing is with your Master's degree. As for how much debt you might have at the Master's level? A state school is going to be much cheaper so look their first. Regardless you need to outline a budget, have a clear graduation plan, reduce your expenses as much as possible, and only borrow exactly what you need.