If using only 120 volts, you have to use a transfer switch designed for 120 volts, not 240 volts.
The neutral and ground are the same with both.
Purchase a 120 volt transfer switch, it only needs a single hot, neutral and ground.
3 wire connection only on a 120 volt transfer switch, not 4..
No way to use an adapter, you do not have 240 volts for the monitoring electronics to use as 240 volt automatic transfer switch.
NOW, if you are using a manual transfer switch, then you just do not use the 2nd input or output lug on the transfer switch.
A manual one will operate the same, it does not monitor the power
CLICK HERE and see one 120 volt transfer switch
While at that site, you can find several more and also manual switches
Let me know any thing further that may be needed,