Take all the costs including utilities and see if you can cover things with a rental inclusive of utilities and still make a profit.

Is the only way to effectively utilize equity to buy the property outright so I would only be paying the interest on the loan leveraged on the equity?

Some utilities stay with the house (eg water if the tenant doesn't pay, you get stuck with the bill), others go with the tenant (eg electricity, phone). 

For some of my rentals, that is included in property taxes, others it is part of the monthly utility bill (water, sewer, garbage) that the tenants pay for directly.

This will further dissuade tenants from using the common areas, which can reduce your liability and risk of fire, etc..If it's a small multi, I would put all the common area utilities on one unit's account and sign a utilities addendum with the tenant and give them a rent credit each month. 

Do any STR tenants pay utilities?

The tenants signed a lease that requires them to pay all Utilities other than trash/recycling.

We utilize a private lender now.

Quote from @Bruno Demarco Quiroz: Hi Amy,When you say “a property that will pay for itself,” do you mean covering HOA fees and utilities, or also the mortgage?

You’ll also want to look at operating costs like cleaning, utilities, and management fees because they’re higher than traditional rentals.