Clarification-
Neither Sony M10 nor A10 provide
phantom power. I suspect you are referring to
plug in power (PIP) which is the typical way of powering smaller 2-wire electret mics into an unbalanced mini-jack input. Both M10 and A10 provide PIP, but I'm not sure what the PIP voltage measures, nor what PIP voltage your mics require to work to specification. The Sony's likely provide somewhere around 2.5V to 3.5V, maybe a bit more or less. The spec may differ between the M10 and A10, but if your binaural mics work straight into the M10, they are likely to work straight into the A10. If not you can always use a small battery box between the mics and recorder, which will typically provide a somewhat higher voltage of around 9V and may improve the the loudness handling capability of the mics somewhat.
Zoom F3 provides
both PIP and [<edit | my thanks to Rairun for the correction]
only phantom power. Phantom power is a completely different powering scheme used for more professional mics that use a balanced connection, typically over a 3-wire XLR connection to each mic. Standard phantom power is 48V and is provided on the two bigger XLR input connections on the F3, while PIP (when provided) is via a stereo mini-jack input. [Edit- however, you can still run an unbalanced PIP mic pair into balanced, phantom power inputs by using phantom power adapters that convert the phantom power to PIP power and the unbalanced mic output to a balanced output into the recorder. They typically look like somewhat longer XLR connectors]
Here's Sony's blurb on the A10's PIP- "When a plug in power type microphone is connected, power is automatically supplied to the microphone from the linear PCM recorder."
https://helpguide.sony.net/icd/pcma10/v1/en2/contents/TP0001668695.htmlHope you can get the M10 working again. If you can't find anyone to take a look at it, maybe you can take it apart and see if cleaning the button contacts helps. I think there are threads here at TS on opening up the M10 and putting it back together again with photos and instructions. If not they can probably be found somewhere on the internet.