That would sound incredibly stiff if actually uttered. אדוני, אסור לרוץ כאן (‘running is prohibited here’) would sound way more natural. נא is something you’d expect to see on a sign and definitely not addressed at a specific person.
This leason is on formal Hebrew- I would expect it to sound somewhat stiff/not the most natural
Is this "adoni" related to "adonai", the word used to replace the tetragrammaton?
Yes, one is literally "my lord", the other, "my lords", because God is plural in Hebrew, (but grammatically singular). Like אלהים, literally "Gods", but you say אלוהים רואה, not אלוהים רואים.