First time home buyer. How do we move forward if we don't know which type of loan we want?

Okay so based on discussions with lenders we can be pre-approved for a $210,000 FHA loan, or a $185,000 USDA loan. We are looking in rural and suburban areas so until we find a specific house we want to make an offer on we won't really know if we can use a USDA loan. In that case, what kind of pre-approval letter should we be taking with us to offers? FHA because it's a higher amount? Both (is that even an option)?

If we take one kind of pre-approval but end up finding a house that better matches the other kind of loan, do we just shop around again for the different loan once our offer is accepted?

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source https://www.reddit.com/r/RealEstate/comments/ika3js/first_time_home_buyer_how_do_we_move_forward_if/