Well I feel really silly. After looking at my image some more I was so baffled at have double connection to the (S) terminal on the compressor. I looked up some manuals from Micro-Airs website and long and behold...
https://pasteboard.co/muahNFRxR0uK.jpgGuess that solves my soft start issues. They have is much different than the manual they sent me though with the wire splicing and such. It seemed like a manual that tried to explain different types of AC units so I thought it was their all in one manual. But this photo of their wiring diagram specifically for the dometic duo therm series shows its different.
Orange wire still went into HERM so thats good. I think their "Blue" line is their white wire (I used grey lol). This seems to be functionally the same, however its backwards and I wonder if that would cause any issues... Essentially my brown wire is connected to a common terminal of the capacitor while my white wire is connected to RUN. This diagram shows brown wire connected to run and white wire connected to common. I will reverse my installation just incase lol.
And last but certainly not least, the black wire was definitely wrong. When I noticed I had 2 wires on S that just didn't seem right to me. Crazy enough, I actually did tie into the black wire originally but I did that AFTER the O/L protection, this shows to do it before. I'm wondering if that is a normally open circuit so it never gets power after the O/L which is why I didn't get the power I needed when connecting the two black wires... I will splice it the way they are showing it here because I think that might be the case.
It doesn't seem like this should damage anything though from what I'm looking at... So hopefully I didn't damage anything lol. Just looks like I ran a double capacitor on the S cycle and nothing on the R cycle so it wasn't able to keep going and tripped.
I bought it on amazon so there was no selections for the model I was installing it on or anything. I guess I just got a generic soft start installation manual. Shame on me for not looking into a more specific manual. But maybe someone else could use this in the future when googling "Micro Air Soft Start Not Working On Dometic AC Unit" lol