A few thoughts come to mind as I read your post. First, while 3 application cycles is a lot and it is expensive there are students that have done more to finally get in. If getting a PhD really is your dream then you can hang in there for at least another round.
Also your comments about the green card make me think you're an international student. I am not an expert on how best to come to the US to study, but I do understand the cost considerations when you have a family. I'm here in California where it is very expensive to live and work/study. Many of the international students/employees I've met chose not to relocate their family because of the cost. No that isn't easy and talking daily via Skype isn't the same. However that was more feasible for them and they just saved up the money to go home during winter and/or summer break. For many of them the separation motivated them to finish faster so they could get home faster. This could be an option for you depending on your family dynamics.
As for the Cornell Masters, I'm guessing you applied to their PhD and they countered by suggesting you consider their MA? That could indicate that they found your application strong, but not quite strong enough to handle PhD work. They may figure that with a bit of polishing in an MA program you'd make an amazing PhD student and they want to be the ones to grab you up and take credit for that. I do not think it is a bad option to consider especially when it will give you the opportunity to study under new professors, expand your research training, and perhaps publish. All of those things will look good to PhD programs. You should definitely ask them about the success rates of students in their MA program going to their PhD program or other PhD programs. Also ask if its possible to talk to current PhD students that were formerly in the MA program.
Now about the unfunded MA and if it is worth it. You'll get a variety of conflicting opinions on this and ultimately you have to decide what is best for your situation and future goals. I went through a couple of PhD application cycles and was unsuccessful. Finally I hit a point where I realized that my efforts to improve my application each year weren't enough to get me where I wanted to go so I needed to do something different. I talked to my LOR writers and other professors and their advice was going the Masters route. Many of the Masters in Psych are unfunded so I created a solid plan to finish on time, work my way through, and cut down on expenses. Is a Masters a guaranteed ticket into a PhD program? No, especially if your desired field is extremely competitive. It does however give you the opportunity to demonstrate PhD readiness, a new batch of LOR writers, broaden existing skills, develop new skills, network, and expand job opportunities. So ultimately I would say that it is a good option for someone who's been unsuccessful at a few application cycles.