Sure sounds like you may have a couple of unrelated problems. Usually, a nuisance tripping GFI is a sign of a worn or aged GFI device. This is assuming there are no down line devices or lights that are leaking small amounts of current to earth ground. This would be uncommon for a light or simple switch, more likely with a motor load on start up or shut down. The more important and potentially dangerous problem is the whole house lighting flickering when the fan and shower light is on. If you have an accurate digital VOM, you can look at the voltage drop at the breaker that controls the bathroom. If you see variations of much more than 5 to 10 volts then isolate which device is causing the fluctuation. (fan or bath lite) by turning them on individually and test again. A older fan is a common source of your problem as moisture and humidity may have invaded the motor and causing larger than average voltage drop to ground. If this fan is in fact the culprit, the fan may be heating up more than normal as well and could become a hazard. If you do not have a VOM or know how to test voltages inside your circuit breaker panel, I'd advise you to seek a licensed electrician soon to correct the situation.
First things first, here's a link to Leviton's 7299 combination switch & GFCI instruction sheet. For a tamper proof it will be a T7299. The only reason I give Leviton is because I know the part number. Hubbell, GE or Cooper are just as good and make the same.
GFCI protection for both outlets.
Follow the instructions that come with the GFCI. There are leads for the switch and lugs for the GFCI and also the feedthru-protection of another receptacle.
No GFCI outlet behind the dishwasher (for easy resetting without dishwasher removal).
This is accomplished by feeding the dishwasher receptacle using the GFCI feedthru-protection lugs.
The switch only toggling the garbage disposal outlet and not the dishwasher outlet.
This is accomplished by using the leads on the combo switch & GFCI to feed the garbage disposal.
Am I going to need to run some more wire through the walls?
If there are no wires between the combo switch & GFCI then you will have to pull some romex between the two.
Should I investigate adding a GFCI breaker for that circuit?
The breaker will cost a lot more than the combo switch & GFCI. Also, if the GFCI trips then you have to go to the breaker to reset it. At least with the GFCI receptacle feeding your dishwasher receptacle, you will be closer.
What other solutions would achieve the same effect as listed above?
I think this is the way to go, so you don't have to pull your dishwasher out to reset the GFCI. The nice thing about this site is some of the people either can think out of the box or have faced this problem before.
Best Answer
A configuration with GFCI protected outlets and switched lights but and unswitched outlets is a completely ordinary and unremarkable configuration.