How to adjust linear voltage regulator equivalent?
You are not thinking about the potentiometer correctly - it is a three pin device with the wiper effectively being the junction of R1 and R2 - you appear to be regarding it as a rheostat i.e. a two pin variable resistor and this of course can be a problem across supplies.

Basically use a pot (maybe 10k ohm) to replace R1 and R2. This gives you a variable voltage between 0V and 30V going into the non-inverting input of the Op-amp.

Also R3 can be a short circuit and R4 is not needed.