I heard because there is HaShem's name in Yehuda that I should pronounce it more like "youda".
does anyone know a source for this practice?
"Yuda" (יודא) is the Aramaic equivalent to the Hebrew name "Yehudah" (יהודה). To my knowledge it has nothing to do with the name Yehudah containing the Tetragrammaton (י''ק ו''ק). It is similar to how you will hear many people say "Yeshua" (ישוע) instead of "Yehoshua" (יהושע). Yeshua is the Aramaic version of the Hebrew name Yehoshua.
To my knowledge, there is absolutely nothing wrong with pronouncing it Yehudah, and not Yuda. It's a matter of preference, how we were taught to speak, and how we talk.
Is this just being stringent?
Well, there's no such requirement. I'm not sure whether it's a stringency.
I know there's no such requirement from the fact that the tetragrammaton isn't pronounced even in the course of public Torah reading, whereas all careful readers I've heard, in many communities over a few decades, including a number of quite learned people, pronounce the ה in יהודה.