I think driver length is one of the most underlooked pieces to a fitting. In theory the longer the driver the further your distance will be.

But if you think about having an efficient swing (smooth, consistent, repeatable, solid contact near the middle of the face, neutral swing path) relative to distance, you can argue a shorter shaft is better.

If you are able to generate faster ball speed/club speed with a shorter shaft, that makes up for the length of driver and if you hit it straighter it's a no brainer. Many tour pro's have shorter than 45" shafts.

Understand you predicament. A fitter would be able to help with this. A shorter length would help you tighten it up a bit. If I get crazy, I will choke up on my driver a bit and get it back in line. Give it a try. I am 6'2", 36" sleeve length as well. Check with you pro where you play.

I took a course on increasing drives. Tips included speed sticks. One of the units was to hit 5 drives each with at least 44 and 45 inch shafts. Maybe a 45.5 as well since that is common across other OEMs.
So with a 45.5 I can get my longest drive. But my average drive distance is better with the 44 inch shaft. Due to hitting the sweet spot more consistently. I have a 44. 5 HZDous shaft in a TSR-2. The closest you can get to is choking down on a 45.5 driver to see if it makes a difference

My TSR fitting this year is the first time we experimented with length and sure enough, the 1 inch shorter gained speed and increased accuracy. I use a +4 grams weight in the head so the shorter length was allowing me to get the heavier head moving faster. Not for everyone but the key is a good personal fitting to find what works for you.

I have tried shorter shaft in driver and it does work but nothing significant that I would stick with it

Also tried a 3 wood with a driver shaft

44.5" - 45" is a good range for the majority of golfers- I would certainly go 45" rather than 45.5"- on a tight coursse especially- I would bet you will get better overall performance at the shorter length.

What happens to a shaft that you shorten which in turn stiffens it then you add the extra weight that you need for swingweight .Does it make the shaft more flexible?

If you add weight it will not make it more flexible but will allow you to feel the head weight a bit more and your swing tempo should be more consistent when all clubs have equal swing weight.

Adjusting length by cutting at the butt does not stiffen the shaft nearly as much as tipping the shaft at the hosel. But the club needs additional weight added to the head with the shorter fulcrum length of the shaft to keep the swing weight the same.
My issue with a TSi-1 with a shorter shaft was not enough weight was added and I couldn’t “feel the head” enough to control the path. A TSR-2 with 4 extra Gmail weight works much better with a shorter shaft.

When you add weight to adjust the swingweight the additional weight actually does change the stiffness of that shaft- shaft stiffness is measured by its frequency (cycles per minute (CPM)- the higher the number of CPMs the stiffer the shaft. When you add weight the CPM measurement decreases so the shaft frequency/stiffness decreases. However the change in stiffness may not be significant- the CPMs decrease by 1/2 of the grams of weight added. For example if you shorten a shaft 1/2 inch and add the six grams required to offset the 3 swing weight point loss from the shorter length, the CMP measurement will decrease by 3 which is 3 tenths of a flex- 10 CPM difference = 1 full flex.. Add 12 grams = 6 tenths of a flex ands so on