my LH level which is getting close to double the top of the range
That is what strikes me as really odd, that your LH is double the normal amount when your total T levels are on the high end and your free T is still considered normal, at least within the normal range, even if it is on the lower-ish end.
When your hypothalamus gland detects insufficient levels of
total T in your body, it sends a message to your pituitary gland to produce LH, which in turn makes your testicles produce testosterone. The conversion of total T to free T is affected (in large part) by your levels of SHBG, which admittedly are high.
I am not going to suggest that I know more than any doctor, and it may very well be that (maybe) TRT will benefit you. But remember that any specialist you consult will see your issues through the lens of his own specialty. Which is not a bad thing, but something important to keep in mind. Like if you go to a dietitian, they'll most likely say it is all due to diet. Go to a psychiatrist, he'll say you some medication. Most of the time, they will not consider the various other causes/solutions to your issue, that is what I have experienced.
And to be clear I am not making any claim at all that your doctor is wrong either, not in the least, but consider this:
I myself have primary hypogonadism (physical failure of the testicles), so when I first got tested my LH and FSH were skyhigh (like 5 times the normal amount). This was to be expected, because my total T levels were around 40 ng/dL (seriously!), so my system was on overdrive and my pituitary gland was putting in "overtime", so to speak. After I brought my T levels to the normal range, the LH+FSH levels dropped drastically, pretty much to 0. This is normal.
But in your case, your brain is producing very high levels of LH despite you having T levels in normal range. It does not make sense that your brain produces such a high amount when your blood test shows totally normal levels.
If I was in your position, I'd make a visit to a neurologist. Even if you don't think that there is any issue with the functioning of your brain, why not go to at least to rule it out? Sometimes, issues connected to the brain (such as a brain tumor) can cause your brain to be in overdrive, even though it does not need to be, like when certain parts of the brain are under constant physical pressure. These type of things happen. An endocrinologist will not be able to diagnose or detect those type of issues, but a neurologist would.
Worst case scenario, you ruled out what might be the most important thing in the world for you to take care of right now.
And the reason I recommend a neurologist specifically is because it appears that your brain is not responding how it should, so there might be an issue there and this might be just one of the earliest signs (in addition to the symptoms you are experiencing on the daily). If it is not related to some malfunction of your brain (let's hope not), then at the very least you can put that suspicion to rest. And if it is, then you can thank your lucky stars you detected it now when whatever it may be is not too far progressed.
Just my two cents. Prevention (or early detection) is a million times better than dealing with it after it is too late.