Anger is a "secondary" emotion; it is used to let us know when we feel someone/something has taken something from us. It is a "motivating" emotion in that it is strong/forceful. As you have noted, you have to "do" something around it (you shut it off).
But it is only the "alarm" that rings when there is a fire, it is not the fire, itself! So, you have gotten in the habit of turning off the alarm instead of looking for and taking care of the fire. Or, to make a more "peaceful" metaphor, it is like an alarm clock and you keep hitting the snooze button and not getting up in the morning; it keeps getting later and later and you know you have to get up so your anxiety rises as it gets later and later. . .
If I were you, I would grab a moment when you feel anxious and look at what thought or activity of yours has made you anxious. When I feel angry, I immediately look to see what is making me anxious, what I feel has been taken from me, what is threatening me. Think of a lion or even a dog and how they snarl/roar or growl/bark? They are feeling threatened.
How I "caught" my anger was in my actions that were not "normal" for me. When I have road rage :-) I know there's something else going on so I immediately pull over if I can and think about what I'm trying to displace. I review my day and/or week. The person in the car I just cut off or who I'm "fighting" with, think is an idiot, or hope they get in an accident down the road, is not the problem! I don't know or care about them.
The anger has to come out, you think you shut it down but you cannot (or you'd burst). You either displace it, where it does not belong (getting upset about inconsequential things; I once called the manager of the grocery store to complain the clerk had put too few items in too many bags so I'd forgotten my $0.43 worth of cat food; what did I expect would happen? How could that help me or the store?) If you think things are other people's "fault" or that the anxiety is "out there", then you are probably displacing anger/your anxiety.
Since you feel you shut down your anger, the anxiety has to get worse and now, with the anxiety med, you have a problem because you are trying to shut down the anxiety! You are playing whack-a-mole with your feelings and that can't work :-)
I would think you may be getting anxious that you are going to be angry. If I wanted to get out of that loop, I would read all I could about anger
What Is Anger? What Is Anger Management? so I felt I understood it better and that the feeling of anger is not about other people's poor responses/actions to feeling angry (abuse, yelling, putting fists through walls or, my favorite that my stepmother did to me; pulling all my clothes out of my dresser and closet and heaping them in the middle of my room floor because I had not put away an outfit like I "should" have; for my brother, she threw all his clothes out his second story window).
Feeling anger can be like hearing an alarm bell so can be kind of scary because suddenly you are in "where's the fire!???!!" mode. Or, you can suddenly remember when your father heard his alarm bell and started taking it out on you so associate anyone feeling alarm bells with alarm/danger to your self instead of being about the situation. You are not on fire, the barn is! The correct response is, "get the horses out!".
So, I would look for whatever is easier for you to spot at this time; either responses you make that are "over the top" in some way, not how you would normally respond or how you believe someone "should" respond or I would look for what scares or makes me anxious.
Ask your T to help point out times when you "should" be angry and discuss those situations in detail/ad nauseam with him/her. Just "accept" you should be angry and work to figure out why, imagine how that might feel, what you get from not feeling and/or where the feeling goes, to squirt out later in the "wrong" place (your anxiety).
Practice "pretending" you are angry about something. Make as many verbal connections between what "should" make you angry and situations you can remember from your past.
"I hate standing in line, it is really annoying and frustrating to me, it is like when my stepmother dropped me off at those strange kid's place and I had to wait all day to see if she would come back for me."
Even if you have to stretch things; there will be more "truth" in whatever memory you grab (your unconscious will be doing the grabbing) than you know at first.
"This clerk is not doing her work, this line goes forever because she's just wandering around and talking to her coworkers instead of helping people. It's like when my stepmother would be sitting, sipping her bourbon and water ordering me around to do chores and not doing anything herself."
At first, you may have to grab any memory at all and it might not seem to "fit". Don't worry about it, your unconscious is working for you and as you get "faster" at grabbing a memory and making a pair and identifying that you are annoyed at the line or the clerk not doing their work or the boss that is making you work overtime or the coworker chatting instead of working, the jerk in front of you driving or the idiot store clerk who can't do simple arithmetic fast or accurately enough, your self will be pleased that you are paying attention and trying and "noticing" so it will show you more and more and make better and better connections and you'll see that it is all about you, not the idiots

around you who have their own sets of problems (the motor vehicles clerk is talking instead of working because her printer is offline, for the 3rd time this week, when are the idiots at headquarters going to get equipment that works or the idiot repair person repair it right and why doesn't her boss adjust the lines, all these people are in hers and they're getting angry at having to wait so long. . .)