When you say transformer, it's more accurately a mains transformer with two 9 volts secondary windings, rectifier (2 diodes) and smoothing capacitors, can't make out if it's a regulated O/P., don't think so.
With the two 9 V windings connected as 9-0-9 and the 2 visable diodes as rectifiers you should get in theory full wave rectified DC of 1.4 times the individual windings AC voltage (off load) if my far distant theory serves me well, ie., 12.6 V. DC..
Regarding the PSU trouble shooting:
What is the DC voltage from the PSU off load?
What current is being drawn from the PSU, ie., is it dropping the voltage?
The 12 V, 2 Amp PSU you are now using may be a higher current rating that the unit in question and is capable of maintaining near to 12 Volts into what may be a faulty stat..
Try disconnecting one stat. at a time and see if the voltage from the original PSU is restored to 12 Volts.
If all that fails you may have a faulty PSU, very unlikely to be the transformer or the rectifier diodes, the most likely components to fail with age as always are the electrolytic capacitors (although they don't look leaky), these should be very easily replaced if necessary.