You are able to save thousands of dollars with the pre-approval because it puts you in a better negotiating position. When you are pre-approved, you are basically a cash buyer. Real Estate Agent making a verbal agreement. Buying a house is a big decision and is a very complex process, so make sure everything is in writing. A lender with lowest interest rate doesn't mean they are the least expensive.
Interest rate is important, but make sure there are no hidden costs.
A reputable mortgage lender will give you a good faith estimate with all costs included, which will usually be the best overall lender. If they won't disclose all costs, stay away. Choosing a lender recommended by a Realtor. Your Realtor is not a loan specialist. They may not know all the loan programs out there, so ask around.