What you describe is more than being introverted, it sounds more like social phobia.
Average people make frequent small mistakes in social interactions and get excused for it. Even big mistakes.
I've improved the greetings that I give to people. Think of the way an stage actor has to speak and act to be seen clearly by the audience. My attempts at "Hi!" were sometimes indistinct so "Good Morning" or "Good Afternoon" became my default.
Practice when the stakes are low. Sometimes when a stranger (server at a pizza by the slice stand) asks "How are you?" I could think "I'm not sure that the really want to know?" or maybe I think I am feeling great and I say and emote "Great!".
When the feeling is real that is a winning move that draws people in. I plant my feelings like seeds.