Hello, looking for some advice about how to use my rental home as collateral for a down payment on a primary residence.

The rental we own is in Missouri, worth about $120k, and we own it free and clear.  We also own a primary residence worth about $260k in Austin, TX area and a vacation home further south worth about $180k, with not much equity in either of those.

We would like to move to a different primary residence because we ran across a home that would work much better for our family (huge backyard for the kids, better location, etc).  I would like to keep the home we live in now as a rental.

The question is, if we want to be able to purchase this new primary residence we need to use the Missouri home by either 1) selling it as quickly as possible and hoping that the house we want doesn't sell, 2) put the home up as collateral for a loan for the new primary residence (I don't know if this is even possible) so we can get financing going faster, or 3) hoping some of you have other creative options that would allow us to make this move pretty quickly.

I know the red flag that's popping up for many of you is the fact that I want to do this quickly.  That aside, does anyone have ideas about how to put an offer in on this new residence and use the equity in my paid-off rental for a down payment, quite possibly before I am able to sell it?  There also may be an issue of debt-to-income ratio if I buy another house.

Hope this makes sense.  Thanks in advance for your help!