Protip for eye contact -- Look between their eyebrows!
It makes it look like you're looking at them in their eyes, but you're actually not!
Just be aware that you should avert your eyes whenever they do, people normally break eye contact after a while during a regular conversation.
I much, much prefer talking while walking as it removes all the weird decisions about what to do with my face and gestures.
My semi-pro tip is to watch their lips instead. I do this because of hearing issues (reading lips helps to fill in the blanks) and it usually doesn't cause questions ... but if it does I just say I need to read their lips.
Looking at someone's lips can also be interpreted as wanting to kiss them, so watch out haha. I read that in a book on body language.