There is a home out there for you. I suggest that you continue to look and be upfront with your situation with potential landlords. Many people are in your situation at this time. Their are plenty of rental homes on the market now. Other suggestions to help: Offer first and last months rent in addition to a security deposit when you sign a lease (if you can do it) . Offer a shorter term lease trial period to prove your are a great tenant then that lease can be extended to full term if all agree. Another thing that helps is if you do not have pets ( more homes are available to family's without pets) landlords generally discourage pets because they are worried about property damage or liability.
All the best,
You may want to take a look at what's available on CraigsList or in a local newspaper. If a property is not listed on the MLS and privately listed you may have a good change at renting a property.