Have you heard any one talk as though they're swallowing, or choking, on certain words?

Yes. I had a co-worker who had all of his teeth pulled, and after he healed up, he got false teeth. He couldn’t hardly talk, much less eat.

Ouch!!
These guys were commentators on TV. They had their teeth.

You may be hearing the received British pronuncisation of the diphthong sch

Schedule pronounced in British Accent

It’s not that excatly, but it may have something to do with the “ch” sound. It happens with words that require them to use the back of their tongue, like the word “school,” with he hard K sound, like “skool.” They seem to swallow the “kool” part and choke on it.
Next time I hear it I’ll mark it and find it online.

FWI I have always thought the British pronounciation of “schedule” should be the right one.

Anyone speaking Portuguese in Portugal. Very guttural language

Girlfriend when she speaks Arabic to her family.
And occasionally to me if she’s annoyed. I understand more than she thinks I do :)