Previous owner of my Excursion didn't have the fobs so I went to the Ford dealer to check it out. They gave me an estimate of $199 for labor and parts. What a rip-off (BTW, haven't been back to that dealer). Since my Ex is a "Limited" I was confident it had remote entry. So I bought these cheap fobs from Amazon for $12/pair and can confirm they work on my 2002 Ex so these should work on your F-250.
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Since your locks cycle when programing the fobs I'm pretty sure that means you have remote entry. You probably checked the fob batteries but if you didn't take a look inside the fobs to see if there's a plastic sheet between the batteries and the terminals. The plastic is to keep batteries from draining during shipping/storing.
The fobs programmed on the first try. They work fine except the range is short maybe 15 feet max. Probably the Chinese batteries need replacing.