Need some help about why this circuit (PSU) isn't working
When you see the 2-3 amps going through the load the lm317 is seeing a current overload situation and it is reacting to it with "Operation In Self Protection" mode. It is noted in the lm317 datasheet in section 8.4.4. 

8.4.4
Operation In Self Protection
When an overload occurs the device will shut down Darlington NPN output stage
or reduce the output current to prevent device damage. The device will automatically reset from the overload. The output may be reduced or alternate
between on and off until the overload is removed.

You might consider putting your voltage varying circuit before the current sensing circuit. Remember your current of the entire circuit is limited to the max current of the ml317. 

You need to combine both voltage and current overload together such that they
both control the 2n3055 transistor. When I find my schematic of such a circuit I will post it.

I did not see a load output of the lm317 circuit. If you are actually putting a short from the right hand 2n3055 that will be your problem.

I found that schematic: