Since you know that the wife's income will no longer be coming when they move in as per their own words, you cannot count on it.
The husband is starting his new job as per his own admission. His current employment income information is moot because it won't be relevant very very shortly. You will have to verify his new job income which would usually with pay stubs. Since he does not have that verify the employment by having the tenant's new job send a letter of employment verification and salary directly to you.
You are not discriminating based on source of income because you are not saying you will not rent to them because there income comes from XYZ source. You are trying to verify the amount of income they will have coming in. If they cannot give you solid evidence (not just what comes out of their mouths) of income, then they haven't met your requirement.
You mention you haven't even checked the credit reports. Pull them. Most likely you can reject them based on something in there such as debit to income, insufficient or derogatory credit score, etc. That would be your best bet.