I've applied for a conventional loan: 3 units, owner occupied, NJ, 5% down.I have enough W2 income to cover the loan and additionally, there are two units rented already. 

I know that going on the loan in my own name would allow me to put more loans under my belt in the future before we get married and become one unit. 

However, you're referring to Fannie Mae's 5% down loan program for OWNER OCCUPIED 2-4 unit properties.

If not - you are bringing cash to closing just to get out of the loans you already have.

So, the seller gets the benefit of charging you a higher interest rate for a loan they did not originate. 

And if you dont actually do that, as @Matthew Paul said, the Baltimore States Attorney was just arrested and convicted of mortgage fraud for not meeting occupancy requirements of a loan for a property on a short term rental.

Legal entities aren't eligible for conforming loans. 

The refinance itself, will be treated as a loan purely to be paid back by the LLC over time.It is best to speak to a CPA or Lawyer pertaining to this first and foremost to properly discuss the private details that should not be shared online. 

Looking for advice on whether I should take out the equity, or seek another type of loan.

If your loan was in fact an FHA loan when you acquired the property, that may be what you're running into.