Okay, I'm not an expert, but I'm keen to learn so some of my suggestions below may be wild goose chases.
The tests below are based on the 9721 datasheet. Since I don't have your multimeter in front of me, you will have to find the corresponding parts. Refer to section 15.1.A Common Use Circuitry (100 pin package IC) for the pinout diagram.
For example, R26 is a 10 Mohm resistor. The 10 Mohm resistor on your pcb may not be numbered R26, but but it will be the resistor that is connected to pin 26 of the IC.
I'm going to break this into two parts. The first will be measuring the volts/ohms jack and the second will be measuring the 10A jack. All measurements can be initially done "in circuit".
Volts/ohms jack
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1) Measure the resitance of PTC (just above the 0.5A fuse). It should be around 1 kohm.
2) Measure the resistance of R26. This resistor is connected to pin 26 (TENM) of the IC.
3) Measure the resistance of R25. This resistor is connected to pin 24 (ONEM) of the IC.
4) Measure the resistance of R24. This resistor is connected to pin 22 (HUNK) of the IC.
5) Measure the resistance of R23. This resistor is connected to pin 20 (TENK) of the IC.
6) Measure the resistance of R22. This resistor is connected to pin 18 (ONEK) of the IC.
7) Measure the DCV of a 1.5V AA battery with the AX-MS8250. It should show 4.15V DC as you wrote earlier.
8 ) Get another multimeter (let's call it DMM-B) and set it to measure DCV.
9) Put DMM-B's black probe on the AX-MS8250 COM jack. Put DMM-B's red probe
on pin 26 of the IC.
10) Put DMM-B's black probe on the AX-MS8250 COM jack. Put DMM-B's red probe on pin 24 of the IC.
11) Put DMM-B's black probe on the AX-MS8250 COM jack. Put DMM-B's red probe on pin 22 of the IC.
10A jack
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12) Measure the resistance of R19. This resistor is connected to pin 13 (SMV) of the IC.
13) Measure the resistance of R31. This resistor is connected to pin 9 (SGND) of the IC.
14) There are four diode next to the 0.5A fuse. Measure each diode using the diode test function using DMM-B. You should have two readings for each diode.
Report your findings using the exact numbered sequence above so it is easy to read. For all resistors, report its color code or SMD markings so we can verify if they are correct.
PS. This thread regarding how Martin killed the UT90C is somewhat related because it uses the same 9721 IC and may be helpful?
http://mjlorton.com/forum/index.php?topic=24.0