Powering Evaluation Board, Voltage Regulator Fizzles Out
What typical current drain when failing?
What heatsinking?

LT1084 datasheet here

Probably but not certainly a heatsinking issue.
You may be able to dissipate 1 to 2 watt in a TO220 package without a heatsink deep-ending on environment.
That's say 150-300 mA max and probably not 300 mA.



Here are some thermal resistance for various package Rthjc (Thermal resistance junction to case) and Rthja (junction to air 0 usually on a 4 layer PCB.
A TO220 package has Rthja of about 30 C/W ona 4 layer board any maybe 60 C/W in air with no heatsink.  

Temperature rise = Watts_dissipated x Rthja in the environment in question. 

Table adapted from here -
MCP1826 - 000 mA, Low Voltage, Low Quiescent Current
LDO Regulator

 

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Power dissipated in regulator ~= Vdropx I = (12v-5V) x I = 7Watts per Amp Or 7 mW per mA. How long it will last deep-ends on heatsinking.
Photo(s) a very good idea. 

This SE answer will probably tell you more than you need or want to know on the subject of regulator heating - My linear voltage regulator is overheating very fast