First things first.....Welcome to the forum. You have come to the right place and I know I speak for all of us when I say that we are glad you are here.
Hmmm....interesting. The gear shift factor causing it does seem odd and makes me think about why. I know mine certainly is affected by the unavoidable "sloshing" of fuel as I drive or stop and is expected to a degree. I know if I want an absolute "accurate" gauge reading....I need to do it when stopped and when the truck is sitting on a flat surface. When I have less than half a tank....if I accelerate quickly from a dead stop....my gauge goes to below E as well.
It may help to understand what/how your "transmitter" (as you called it) actually works. In simple terms, it operates on the same principle that a strain gauge operates. For those who may need a refresher on what a strain gauge actually is....follow this link:
Strain gauge - Wikipedia, the free encyclopedia
As you can imagine after reading about what a strain gauge is....it should help explain why they tend to fluctuate. GM fuel gauge senders and the gauges operate on the range of 0-90 ohms. 0 ohms indicates "E" and 90 ohms indicates "F". Other manufacturing companies use different ranges, but this has been GM's standard range for many years. If you can imagine this....a disconnect on the wiring from the sender (or transmitter) to the gauge will send an "infinite" resistance reading and would cause the gauge to read way past full (as far as the needle can physically go depending on design). For the gauge to erroneously read a lower level than it really is requires a lower resistance to be sent. It usually isn't the case and I would imagine that a bad sender either sending a lower resistance than it should be....or a bad gauge displaying a level that isn't reflective of the ohms it's being sent are the only two instances I can think of right now for one to see a lower level than it should be.
Back about 12 years ago, both me and my girlfriend at the time were victims of a bad batch of gas from a local refinery in south Louisiana. I had a 98 C1500 and she had a 99 Tahoe. What it ended up doing was causing your fuel level to show about 1/3-1/2 lower than actual. The cause ended up being that the bad batch had a chemical in it that would severely corrode the sending units in fuel tanks. Especially GM because of the materials they used for the strain gauge in their senders. We both were issued a check for about $400 bucks to pay for the damage caused by this. My girlfriend wanted hers fixed and that amount covered the local dealership to drop the tank and put in a new sending unit (and solved the problem). I just kept my $400 bucks and learned to live with it and simply watch the odometer. I used the $400 to pay for a Line-X bedliner instead. Anyway....just thought I would share that to help explain what I was trying to say.
I'm a little unsure what your gauge is actually doing and am not sure if you are talking about level fluctuations or if you are talking about the gauge just stops working completely (goes to E) and suddenly starts working again. If that is the case....then it sounds like you have a loose connection somewhere on the power going to the gauge. The gear shifter helps it make the connection and allow the gauge to turn back on. Just depends on if you are seeing a fuel gauge level issue or fuel gauge working at all issue.
hope this helps.