'Say hello to him' or 'Say thank you to the man' is awkward?
Please note that I wrote "_can_ sound awkward" no "inevitably sounds awkward". Yes there are times when "say something 'to someone'" are okay, but there are times when it's not.
Also note that neither of these violates the 'say something' rule. Adding 'to ....' is more an afterthought and the sentences are fine without them
Say hello. Say thank you.
But unless you're carefull you'll open to the door to:
"He said to me to leave. " or "I'll say it to him" or "I'll say it to you again." which are understandable but awkward or at least not optimal.
Again remembering the rule that say has to include a reference to what was said while tell has to include who was told is a good basic rule. Refinements (like when you can add a 'to someone' phrase come later.
adding this due to the stupid posting policy:
Another pitfall for Poles is 'How can you tell?' which they often translate as 'Jak tak możesz mówić?'
I would just treat that as a separate verb that happens to be written and spelled the same. Or, if the students are up to it, a separate use of the verb unrelated to speech acts.
That is, a secondary meaning of 'tell' and arguably 'say' is (roughly) wiedzieć but this is limited mostly to set expressions.
I can tell. (Wiem i tyle. Znam się na tym)
It's hard to say. (Trudno powiedzieć. Nie wiadomo.)
One of the hardest things about English is the difference between productive meanings that can be freely used to generate new sentences and set expressions that have to be treated separately and learned one by one and which have no wider application.
Even native speakers (I don't exempt myself from this) can get caught up and forget about the latter.