The OKI SA-1000 power transformer A41-80-28A33 is made by Pulse 28VCT 2.8A or 14VAC 5.6A 80VA.
I would expect 22-19VDC after a fullwave CT (2-diode) rectifier and filter cap. You are getting more than spec because there is no load on the transformer and your mains voltage may be high.
T12 clone controllers input is 24VDC 3A peak. There is a 30V limit - the MOSFET and sometimes 3.3V regulator is rated for that max. It depends on the version and make. They do fine on 22-19V. So the transformer would work fine, add a rectifier and big filter capacitor 3,300-6,800uF 35-50V to power the T12 controller. T12's suck for a decent stand though.
It's possible to draw a schematic of the OKI controller and see if it can be modded to work with a Hakko 938/888 wand.