Okay, I have a couple additional ideas:
#1:
You may have to have 2 separate save actions. The first save action would need to save the “New_Available_Billings” Model. The second save action should save the “New_Incoming_Payment_Applications” Model.
The Billing Record has to be created before we can save the payment application.
#2 (longshot, but have to ask):
Are there any conditions on the “Billing__c” field on the payment applications model?