Your 23VA transformer may not be sufficient. When calculating the permissible secondary I
RMS for a transformer, you can't use the DC output voltage after the regulator, or even the DC voltage before the regulator, but must use the nominal AC RMS output voltage. If the transformer has multiple identical secondaries, the calculated current must be divided equally between them.
Then there is a topology dependent derating factor going from from I
RMS to I
DC to consider. For a bridge rectifier directly feeding a reservoir capacitor, that factor
* is 0.62, so for 1A @12V DC out you need approx 1.6A RMS.
The regulator requires headroom to operate, typically at least 2V for a non LDO regulator, and the input ripple trough must be at least that much above the output voltage. If you do the math (or crunch the numbers in a SPICE sim), by the time you've allowed for diode drop in the rectifier, and assuming reasonable size reservoir capacitors you need something like a 13.5V secondary to ensure it can still deliver 12V at full load during low line conditions (i.e. when the mains supply is at its minimum permitted voltage).
Therefore *IF* your 23VA transformer is suitable for a 12V PSU (and it may not be if its secondary voltage is too low or too high) at best, assuming a 13.5V nominal secondary voltage, it will be good for 23/13.5 * 0.62 A DC = 1.06A TOTAL. If the secondary voltage is higher the limit will be reduced proportionately. If the voltage is lower, there will be an increasing tendency to drop out of regulation at full load resulting in ripple breakthrough.
If the nominal secondary voltage isn't marked, measure its secondary voltage unloaded, and in the absence of any other data, assume 10% regulation, so take the nominal voltage as 90% of the measured voltage, and recalculate the max continuous DC current accordingly. N.B. line frequency transformers tolerate short term overloads very well for minutes at a time, so if it doen't quite meet the peak current requirement for your design it may not be problem as long as the average current is well under the calculated limit.
* Hammond Transformer division's Design Guide for Rectifier Use gives derating factors from IRMS to IDC for various transformer secondary, rectifier and filter capacitor typologies. Save a copy and study it!