Kyle, I think that's progress!
Tightening that battery cable made a difference, albeit small. So I think that brings us back to ground connections or, perhaps, a battery condition issue. All the grounds need to be clean, bright & tight. The critical one is under the right side engine mount, but there are several others. I think that if the speedo is affected by the headlights, it's providing a missing ground on those circuits, if you follow.
I'd clean all the grounds you can find!
With or without the sensors connected, can you try turning the traction control/DSC/ASC off & see if anything changes? (Hold the switch down for 5 seconds).

As for the sensors, I'd talk to your friends at AutoZone & see if they can explain why they list so many for the Mini at so many different prices. I'd have thought Dorman to be a good brand.

The battery date is stamped into the top of the ground terminal post. NN YY where NN is the week# & YY is the last 2 of the year.

 

+1 one on having the battery Load Tested, the best way to make sure.

I just read the thread and it is mostly the same things I just posted in another thread.

MINIs hate low voltage.

Best of luck.

Motor On!

 

I'll be getting my battery tested tomorrow. I'll let you guys know what happens

 

Went to my buddy's shop, he plugged my car into his computer there. Both front and left rear speed sensors need to be replaced. I guess I installed a faulty speed sensor. He also told me that's why my speedo was acting up.

 

Thank you to all who replied and read my thread. I appreciate you all.