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.
If you have bypassed the GFCI receptacle and still have problems, then there must be a loose connection between the GFCI and the breaker. Here are the places to look:
At the GFCI, are the wires connected by a screw, or by push-in? The push-in connections are unreliable - I have had a receptacle "fry" because of poor connections, and it was installed by an electrician. Installers like them because they are fast, but I always go with the screw connection.
At the breaker, is the breaker making good contact with the power bus? If the bus is corroded, even a new breaker might not make good contact. Try plugging the breaker into a different slot, if there is one to spare.
Also in the panel, is the neutral (white) wire screwed down tightly?
Are there any other outlets between the panel and the GFCI receptacle? If so, make sure the twist-on connectors are tight, and that the splice is not made through the receptacle.
Note that loose connections will not trip a GFCI. Neither will short-circuits between hot and neutral - they will trip the breaker in the panel. Also, nothing "upstream" will trip a GFCI. And once a GFCI is tripped, it won't reset itself.
Best Answer
Is this a new outlet where there was not a GFCI before?
Was the unit working before and suddenly stopped working?
There seem to be too many downstream devices on the load side of the GFCI. I would have expected this list of devices to be on at least 3 circuits. That may explain why there still live circuits after shutoff to the GFCI.
In general, GFCIs won't reset when
To isolate the GFCI from any downstream problems, disconnect the load terminals and see if it will reset.