Without seeing the paperwork, I think this is where you went wrong:

"New vehicle: Cost $31,435 (which includes $2,759 negative equity)" <== No; only Your Cost (basis) goes here

"New loan: $31,435" <== Yes, you are More in debt than the new Vehicle cost, even after your down payment

New car = $28,676 Your Cost?

Like this:

I bought a $5,000 vehicle but am in Debt $7,000, because I essentially refinanced the negative $2k equity

And all of this assumes there is no Step up in basis or "boot" involved; always review this type of activity with your own CPA or tax preparer.