So another update!
Decided when I got home, as I wait for parts to arrive, that I will double check all the connections and grind away any rust and all that type of stuff...
Anyway, after all that, I turned the unit on. Same as before... However, the "relay" was making electrical arcing sounds and really sounded like it was struggling. After another 10 seconds or so, the (transformer?) or switch dropped and the unit kicked on... Fan, compressor, etc... For about 5 seconds. Then it shut off and went back to humming. Did it once more for a few seconds and then I shut it down.
Looks like it's the contractor that failed (had to look up the name!)... If everything else kicked on, it's a good sign. So I removed the contractor and went off to Home Depot but no luck... So resorted to Amazon for a replacement part for $8 and will have it Sat/Sun... So fingers crossed... I may be back in service if this goes well.