What's likely happening is that your debt to income ratio is being thrown off by the possibility of renting. For example, you say your current payment is $623. Let's suppose, to round numbers off, you can get $600 in rent. Nearly a break-even, right? Not from the lender's perspective. The lender usually will take some percentage of the rent, often 75%, and credit only that portion to your income. (They figure you'll have some vacancies, some repairs, some other expenses.) So they'll discount that $600 in income down to $450. Now you're showing a negative cash flow, from the bank's perspective, of $173 a month. And, obviously, if you can't get $600 a month in rent, your negative will be even higher.
I'm not a lawyer or an accountant, so this isn't legal or accounting advice. First, though, you should check with an accountant. And closely examine your own finances to see if you can afford to rent.
There are a couple of ways you might be able to get "around this." First, if you are willing to sell your house, but have had trouble doing so, or figure the time isn't right, you might offer your house to someone on a lease-option. Lease the property, giving the tenant-buyer the option to purchase for a specific amount at some point in the future. Generally with lease-options you can charge more than straight rent; the additional is credited to the purchase price if the tenant-buyer purchases. So, taking our numbers above, if the fair market rent is $600 a month, you might charge $700, crediting $200 a month toward the purchase price. That's just boosted your monthly income by $100. And it's also made the house attractive to a buyer, since their effective cost is really just $500 a month ($700 in rent minus $200 in option credit if they purchase).
Another way around it is to transfer ownership of the property into an Illinois-style land trust. The new owner of the property will be the trustee, though you'll retain power of direction over the trustee. The mortgage will still be in your name. However, the trust will show up as an asset--as personal property--on your tax forms and other financial statements. And you'll arrange for the trust to make your monthly mortgage payments. The monthly $600 from the trust may not be discounted by the 25% or so that the $600 from a tenant would be. Again, check with an accountant on this.
Other, more conventional options, would involve finding another mortgage company, putting more down on the new house, or finding additional income (for instance, through a part-time job).
Hope that helps.