Mike,
If I understand correctly, you have a power supply with a transformer that has a secondary voltage of 38VAC.
As noted above, rectifying 38V will give you a DC voltage of more than 50V.
Linear regulators will be hopeless at dropping the output voltage down to 13.8V. You will waste three times as much power as you use in your load.
A switchmode buck converter is probably the best way here, although the complications of rolling your own are considerable enough to challenge the inexperienced. It may be possible to find a kitset that will suit.
Back in the old days before cheap switching electronics, the solution would have been to place an SCR in the primary circuit to regulate the output of the PSU by chopping in to the mains waveform in the same manner as a dimmer does. This needed fearsome output filtering to eliminate the harmonics, but it was the standard fare for large rectifier sets such as were used in telephone exchanges.
Best of luck,
Mark Monson
