Tinnitus is really complex.
I don't think headphones will make tinnitus worse when you use them at low to moderate volumes. If you crank them up, it could damage your hearing and that can make tinnitus worse. But as you said, blocking out the noise will make the tinnitus more apparent, which could make you focus on it more, and that can make it sound louder.
I have had T for over 20 years, likely from playing drums in rock bands and going to a lot of shows without earplugs. I started using hearing protection when I started to notice it, and it largely stayed the same until recently, only noticeable at night when it's quiet.
Two years ago I had a spike in T when I moved to a quieter house and had a lot of stress. I struggled with it for a couple months but then found that meditating helped quite a bit by lowering my stress levels, and it went back to previous levels. Since then I used headphones a ton, closed backs and ANC headphones, and never had issues.
About six weeks ago, I went to a concert and noticed by the end of it that my ears were really bothering me, and afterwards I had some muffled hearing and ringing, and that has largely not gone away. I've seen an ENT and waiting for a hearing test to see if I lost any high end hearing. It seems worse than it did two years ago, and meditation has not helped like it did previously.
One of my issues is that I am prone to anxiety, which is highly linked to T and can make it seem much worse. The more you focus on it and worry about it, the more your brain amplifies it. It got really bad about two weeks in and I saw a GP to get some anti-anxiety medication. That has helped quite a bit, reducing most of the anxiety and thinking, helping me to sleep a little more, and reducing the T a tiny bit.
I've watched a lot of videos on the topic, and there's some paths for improvement. There's no cure for damaged hair cells (at the moment), but the ringing is something your brain is doing, not your ears. There's a process called "habituation" where your brain adjusts to the ringing and tunes it out, which is generally how I've lived the last 20 years. The key is to minimize anxiety and reduce stress as much as possible, then to start forming new neural pathways in your brain with positive thoughts and feelings. I haven't fully figured that out yet but that helped me a couple years ago, and I'm working on it again now.
I do hold out hope that I'll get back to the levels I was at before, and I've heard and read a number of people who practically eliminated tinnitus. Gotta stay positive.