Linear regulators are ok but wasteful of energy (heat) and often fairly large due to heatsink requirements and the use of standard iron-cored transformers (yours is some 120W, conservatively and the transformer would be appreciatively large). Switched-mode regulators are available that have adjustable output voltages. You might consider a small SMPS PC power unit - more so if you can trace the output circuitry to isolate the neutral (negative) lines whereby the 12V and 5V outputs could be wired in series to give you the 'nominal' 18V required.
I am assuming the voltages you've specified are of the usual +/- 10% requirements?