I've been taking psychedelics for about 15 years and dealing with generalized anxiety disorder and panic attacks for about 10. I never really had anxiety when tripping before the anxiety disorders became a thing, though. I've found that some psychedelics have more of a tendency to induce anxiety than others, and for varying reasons.
DXM will sometimes make me pretty anxious during the come up and come down, but rarely during the peak. It's mostly from the physical discomfort of those periods for me. I've found that going for a long walk and listening to headphones can be a really nice distraction during these periods, and as an added bonus I usually wind up finding some relatively private (for being outside) spot to relax in for awhile once I really start tripping. I think I'm just too detached from things to have much anxiety during the peak, and since physical sensations are generally pretty absent at that time I don't stress about much (my anxiety usually being triggered by worries about my body). I don't know if you're a pot smoker, but getting stoned while tripping on DXM pretty much ALWAYS gives me a panic attack, and I've heard similar reports from others. Unfortunately with DXM there's not a lot you can do aside from try to relax and ride it out since most anti-anxiety drugs are fairly unsafe to use in combination with DXM.
I've found with mushrooms that as long as I'm with a good friend who will be understanding and caring if things go south I can pretty easily deal with most anxiety. What I find triggering with mushrooms is just how emotional and confusing the experience can be, it puts me in a very vulnerable position which can be very bad if I'm around people who are going to roll their eyes anytime I express anything other than blissed out wildness. Being alone on anything other than a very light dose can also be very anxiety inducing for me I think because the intensity of emotion is just there inside of me with nobody to talk through it with or relate. On the plus side, with mushrooms it's generally pretty safe to take most drugs to treat acute anxiety if you have access to them, albeit they might dull the whole trip. I'll take a slightly duller trip over an hours long panic attack any day, though.
On that note I would just say that you pretty much just have to try and develop tools for dealing with it, and make sure your environment and trip buddies will be good for you. Of course you can only really know what works for you through trial and error, and it is entirely possible that these sort of substances just aren't for you at this point. Maybe they will be later, maybe not. I'm curious whether you deal with anxiety in general outside of drug use, or if it only rears its head in those instances. If it's the former, I would suggest maybe focusing on getting a handle on that before you dive in with psychedelics, or at least stick to low doses and maybe build to more powerful experiences if those work out for you.
Good luck!