Hello, and thank you for your question. Have you considered am FHA home loan? FHA guidelines state that:
"Part-time and seasonal income can be used to qualify the borrower if the lender documents that the borrower has worked the part-time job uninterrupted for the past two years, and plans to continue. Many low and moderate income families rely on part-time and seasonal income for day to day needs, and lenders should not restrict consideration of such income when qualifying these borrowers.
Part-time income received for less than two years may be included as effective income, provided that the lender justifies and documents that the income is likely to continue."
In your case, you can justify your income because you have a 3-year history of making ends meet with part-time income, and because your new position provides an increase in pay.
If your employer is willing to complete a Verification of Employment (VOE) form, that could be very helpful. Here is a link to a sample form. Line 11 asks for "Probability of continued employment," and if that response is positive, that would go a long way toward reassuring the lender that your income qualifies under FHA guidelines.
Note that you can't give the form to your employer yourself; the lender would have to do that. But if you know in advance what they would say, ask the lender to use the form instead of just looking at your W-2s and pay stubs.
Finally, you probably qualify for some form of down payment / closing cost assistance. Income limits are higher than most people know. The Mortgage Reports has a complete guide on Down payment assistance in the USA that would be useful for you.
Good luck and please follow up if you have additional questions.