The best thing for you and your husband to do is get yourselves a buyer's agent. You need an agent working for YOU and YOUR Best Interest. Remember the listing agent's job is to sale the home in the shortest amount of time and for the highest possible price. They owe that to the seller. The buyer's agent's job is to find the buyer a home that fits the buyer's needs and get it at a price that is RIGHT for the buyer.
Now for the cross approving,many times a seller/listing agent will require the buyer to be cross approved by their preferred lender. I would not call this a scam, it is an attempt to make sure the buyer is REALLY qualified. It is can also be an attempt in my humble opinion to get business for their preferred lender. After all if their lender cross approves you and offers you a better rate and terms than the lender you chose to be pre approved with, you may decide to work with their lender.
I have a client who has pre approval from Wells Farge, and Bank Of America, and I was told she would have to get cross approved by a loan broker the seller/ seller's sgent preferred.
You would think Wells Fargo and Bank of America would know if she is qualified or not.
Now, as for homes being listed for sale that have actually sold. That does happen.Many consumer based websites simply to do always have up to date info. If you really want to buy a home and have access to properties that are available for sale yo MUSt have a buyer's agent searching the MLS for you. Keep in mind homes are moving very fast these days. It is not uncommon for a home to list on a Weds, have multiple offers by Friday, and be in escrow by the following Monday.
At 350K you should be able to find a place to call home, you just need yourselves a good buyer's agent.
Best of Luck to You!!!
Kawain Payne, Realtor