A couple of suggestions, having once been an anxiety monger myself.
Don't let yourself be too delayed over who to see first. Just schedule all of what you need as you reasonably can. Get the physical, get an appointment with a psychiatrist to talk about medication for anxiety, and get an appointment with a psychologist/therapist for some talk therapy. In any order, as you can, all of those are useful.
The difference between a psychologist/therapist and a Psychiatrist is that the later is an actual MD who can prescribe drugs, but most are not focused on extensive talk therapy once they understand the details well enough to prescribe. A psychologist or therapist are usually about talking through how you're coping with things and working out ways to express yourself better, handle your feelings better, etc.
As for what drugs are out there, too many to list them all sanely right here. My suggestion is that you talk to the psychiatrist and ask him the names of the drugs he is most interested in having you try, and why. And ask him to hold off sending it to the pharmacy for a day while you research them to make yourself comfortable. Confirm that it's okay to call the next day to get a script once you're comfortable starting one of them.
As for what to say, I find that starting out with something you wrote can help break the ice. I would take approximately what you said here in and just hand it to the psychiatrist and psychologists appointments. It gives you a place to start and they can then ask you some questions. It will be hard at first. It's okay, if it helps, to ask someone to go and hold your hand, if you feel comfortable talking honestly in front of them while they're there.
For me it works best just to start with saying "This is incredibly hard for me." And then proceed from there. They have some sense of that already with most people, but it makes it easier inside my head to preface it that way so they are likely to be gentle with me until I can handle more.
If you can journal, consider doing so about the feelings as they come up. One thing I did with my therapist was to have two diaries and each week I went, I gave her the one I had been writing in last week and she gave me back the one she'd been reading from the week before. It let me cover a lot more ground with her very quickly and let her come in with some questions about things she wanted to make sure I was working through from last week, each time.
You can get through this. It's very hard, but you're a remarkable person for tackling it and staying brave even through the tears!
Also, make sure you're eating a lot of uncooked plant material daily. It really made a difference for me to switch to organics and remove high flash point oils from my diet, stop consuming soda and High fructose corn syrup and un-natural additives and preservatives. And I stopped eating much if any fast food. It made a dramatic difference in my anxiety and depression and I was able to get off the meds for those problems because my system was regulating itself so much better when not bogged down with garbage.