I am happy to hear you figured out your problem. For anyone else reading this, I should point out that the three batteries in series make it a 36 V system, not a 36 watt system (that will depend on the current).
Just changing the batteries for higher capacity ones shouldn't cause your motor to die. However, there might be other factors like the new batteries being heavier, which could put more strain on your motor and lead to the failure.
You also wrote "batteries" (plural); if you originally had something like two in parallel and changed it to two in series, you would have gone from 12 V to 24 V, and that could definitely lead to your motor wearing out quickly.
A final thought I have is that the motor is not necessarily what broke; have you verified that independently? The things like higher load and higher voltage that could damage your motor could also damage the control electronics.
As long as you are talking about "amps per hour", you do not understand it. "Per" means you are doing division, so something like "watts per hour" is very different from "watt hours", which is watts *times* hours. Units like amps and watts are already rates (coulombs per second and joules per second, respectively).
As for your main question, I specifically went over it in the post, and you got it right in your follow up post, except for the "per hour" part: a 2Ah, 10C battery can deliver 20 amps.
We have what should be a final configuration for the corresponding sensor board, so we just have to do initial production and make the product web pages and documentation. We should have that done by the end of the week.
How well different battery types handle high loads is generally a very relevant consideration, but your ~2 A discharge rate is not that much for these batteries, so the alkalines having 5x the capacity should far outweigh the variations in high-discharge rate characteristics.
This is not that complicated, and your latest question is whether 26 is bigger than 17. If you gave two more similar numbers, like 17 and 19, you should just try it and time it if it matters much if one does 8 hours and the other does 8.5.
Thanks for that data point. I just did a quick search on Hitec's site (www.hitecrcd.com), and I see no claim there of the HSR-1425CR being digital. They say it is the continuous rotation version of their HS-425HB, which doesn't lead to anything in their search results, but HS-425BB is listed under analog servos.
If the servo is analog, the waveforms showing current in this article are consistent with your observation that the servo gives you more power out with a higher pulse rate. If Hitec is not documenting this, I think there is not much value in doing further characterizations since we do not know how much the servos can vary from unit to unit.
You are correct about the calculation for the four hour estimate, and I think the "12AH/20HR" means that the 12 Ah capacity is based on a 20-hour discharge (i.e. discharged at 12/20 A, or 600 mA). In general, you will get less than the 12 Ah out of the battery if you discharge it faster, but you will probably still get close to the four hours.