I'm very sorry to see you are suffering like this. I'm in a similar boat and I may share some things I learned along the way.
I found L-theanine would worsen my tinnitus. It's supposed to be gaba-enhancing but it can be glutamate-enhancing. Not necessarily your case but be careful. Perhaps taurine could be another choice to help gaba.
From:
https://www.holistichelp.net/blog/how-to-increase-gaba-and-balance-glutamate/
If you check that article, it presents a very nuanced analysis of how one can try to increase gaba in the brain (and hence calm down, fight anxiety and possibly quieten tinnitus). It is long but perhaps worth reading if you wish to go the supplements route.
For the private psychiatrist, I hesitate to suggest drugs to investigate with him as these are all double edged swords. There is a famous protocol by Shulman et al based on brain imaging, this is clonazepam plus gabapentin and it seems to work for some with severe cases BUT both drugs are very addictive, may have bad side effects and you would take a very big risk of dependence, and they may also turn on you after a while,
@Star64 is an expert on how these drugs can go wrong and she is great. Of course if you are about to jump off a bridge the cost/benefit analysis is rather stringent.
@linearb is an authority on taking these drugs here as he has done a lot of research and is using them. It's a cost/benefit analysis you might discuss with your psychiatrist but keep in mind a non-negligible amount of people get tinnitus when they discontinue these drugs. It's really desperate measures and if possible at all they should be avoided. Keep also in mind that psychiatrist constantly under-estimate the impact of psychopharmaceuticals on tinnitus and might tell you that some are safe when they are not.
Other drugs that have been shown to work in some studies but failed to replicate success in others and have only limited support or anecdotal evidence (as in failed trials etc) are:
1) acamprosate,
2) neramexane,
3) sulpiride and melatonin, 4) pramipexole,
5) Caroverine, 6) Flupirtine, 7) Memantine.
There are also potassium channel openers, but retigabine has been removed from the market.
In terms of antidepressants, be careful as there is research showing that serotonin may increase tinnitus, so serotonergic antidepressants are risky. People get tinnitus from SSRI withdrawal or even while under SSRI. Despite this, there is a study showing that Sertraline works for tinnitus but again it's risky. Also, if you would like to go serotonergic perhaps natural supplements like 5HTP, tryptophan or St John's Wort are worth considering but never take them together with serotonergic antidepressants to avoid serotonin syndrome.
What else... some people have luck with tryciclic anti-depressants like Amitriptyline (but again serotonin...) or Noritriptyline but you hear all sorts with these two.
A modern antidepressant some will propose is Vortioxetine, it is serotonergic partly as a modulator if I understand correctly. It does not seem to help much with tinnitus from what I heard.
Agomelatine is another antidepressant that acts on melatonine, it's considered to be safe and might help with sleep. I was suggested this in the past but it had bad side effects with me.
For sleep they might also prescribe you antipsychotics like Quetiapine. Be careful again about side effects and discontinuation, it can be quite tough with these drugs.
I would like to give you an easy answer but it's not there, it's a jungle. Perhaps the easy thing is to ask you psychiatrist to make suggestions but making him aware of the risks with tinnitus. Perhaps show him the research piece where they show that serotonin may increase tinnitus.