I think you should try some Darlington transistors in there.
This is based on the idea that, if changing the capacitor size isn't allowing enough current, you must have run out of current through the transistors.

 

The weird thing is that if the transistors are isolated from each other then they will do their respective job of High or Low.

When I connect their bases together and their emitters together then it doesn't work.

 

You only need one capacitor since, from an AC viewpoint, the two are electrically in parallel.

Try one 470uF capacitor to ground.

 

Do you know why I can't get it to switch back and forth using the Odroid?

Because the 3.3V output can't drive the base of Q1 high enough.
You could try the following circuit mod, but with only a 5.6V supply you're pushing your luck. The downside is that there would be a standby 5mA current if the Odroid output stayed at 3.3V. So make sure its output is kept low whenever possible.


Edit: Adopting Crutschow's suggestion, the top cap C1 can be 'borrowed' and paralleled with C2 to effectively double its value, like this:-

 

Thanks guys for the suggestions. I'll give all those a try.

 

The downside is that there would be a standby 5mA current if the Odroid output stayed at 3.3V. So make sure its output is kept low whenever possible.

Are you saying that 5ma would be drawn down from the 5.6v battery and not from the Odroid?

 

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Edit: Adopting Crutschow's suggestion, the top cap C1 can be 'borrowed' and paralleled with C2 to effectively double its value, like this:-
View attachment 81967

R1 should be about 4.6kkΩ not 10kΩ to give a base current of 1/10 of the collector current.

 

Are you saying that 5ma would be drawn down from the 5.6v battery and not from the Odroid?

Yes.

 

I could not get the Darlington transistors to work or the 470uf capacitor by itself to work. The extra transistor and resistor did work with either the two capacitors in series or the one by itself.

I did notice almost 5ma standby current. Could I get another pin out of the Odroid to supply this to the positive rail so it wouldn't drain the batteries?

Out of curiosity I tried a different relay and a Raspberry Pi (higher current) and could get the attached circuit mod to work and turn both on or off.

I have a basic IC question I can't find the answer to. On the output of the Schmitt inverter I measured about 150ma or so current for the one pin that I am using. I think the datasheet lists 24ma. I'm not using the other 6 outputs. Is each pin limited to 24ma? Or if you are using only one pin like in my example can you get 24ma x 6? Or am I just making it put out more current and thereby damaging the IC?