Hi Marshall,
You should not need a 'new license' - just because you have chnaged your hardware. In fact, you can use PixInsight on both your old and new machines (and any others, Linux or otherwise, that you might have) all at the same time.
What you need is the license file from your initial install - which you can then copy into the same (equivalent) directory folder on any and all other machnes that you want to run PixInsight on.
Do you still have the hard-drive contents of your original machine?