Does my duplex carry itself once I move out?

Less about the loan and more about the unit you're vacating: have you actually priced what your side rents for, or are you assuming it matches the other side because the layouts match? The unit somebody has been living in as a home usually needs work before it shows like a rental. Paint, whatever you did to the walls, the appliances you were happy with because they were yours.

I'd separate this into two questions because they have different answers.

Qualifying: most conventional underwriting will count rental income from a departing residence, but it generally wants a lease and often proof the deposit cleared, and it applies a vacancy factor to the gross. So plan on roughly 1,950 counted against your 2,180 duplex payment. That leaves you carrying a small negative on the duplex plus the full 2,600 on the new house, which is where your file gets tight, not the down payment. Ask your lender to run it both ways before you do anything irreversible.

Cash: 5 percent on 340k is 17k, and closing on a purchase that size is realistically several thousand more, so you're looking at 22k to 24k out of 31k. Then you have a turnover on your old unit at the same moment. Two months of both payments with no rent is 9,500. Your 31k does not cover the down payment, closing, the turn, and a gap.

So I wouldn't sign a lease on your unit first without somewhere to live lined up, and I wouldn't try to close a purchase without the lease if your lender needs it to qualify you. The way out of that loop is usually the sequence where you get the house under contract with a longer close, sign the lease on your unit to start after your closing date, and move once. That only works if the lender will accept a lease that starts in the future, and some will not. That's the question I'd take to them tomorrow.

Otherwise you wait a year, get the duplex rent documented and your reserves closer to 45k, and buy in a stronger position. There is nothing wrong with the pace being one property every two years instead of one every year.

The mechanism arbor described is the one that governs your file. When a borrower converts a departing primary residence to a rental, underwriting normally requires documentation of the new rental income before it can offset the existing housing payment, and it discounts gross rent to account for vacancy and maintenance. The size of that discount, what documents satisfy it, and whether a lease with a future start date is acceptable vary by loan program and by individual lender overlays, so get your specific lender's requirement in writing rather than working from what any of us describe.

On the numbers as posted: 2,600 gross rent on the duplex against 2,180 PITI is a 420 spread before any discount, and 1,950 counted against 2,180 is a 230 monthly negative added to your obligations. Combined with a 2,600 payment on the new house, your housing-related debt service before the rent offset is 4,780 on 7,667 gross monthly income. Whether that clears depends entirely on how much of the 2,600 rent the underwriter allows and what other debt you carry, which you have not listed.

The assumption doing the most work in your post is that your unit rents for 1,300. That figure comes from the other side of the building, leased at some point in the past under whatever condition it was in. Pull current listings for two bed one bath units in your immediate area and price your side on today's market and today's condition. If the real number is 1,150 and the turn costs 4,000, the entire plan changes shape.

Reserves are the other constraint. Lenders often require reserves per financed property, and that requirement grows as the portfolio grows.