Full-back Watson, 26, made 30 appearances for MK Dons in the third tier last season before opting to leave the club following their relegation.
Hatters man Watson, 22, featured nine times in the 2022-23 season.
A Republic of Ireland Under-21 international, he moved to Kenilworth Road after two years at Derby County.
The pair become Charlton's eighth and ninth signings of the summer transfer window.
"Tennai is a young defender with all the attributes we are looking for in a full-back," technical director Andy Scott told the club website.
"We have had a number of injuries in our midfield area and so the acquisition of Louie will complement the group well and give us more strength in depth."
Find all the latest football transfers on our dedicated page.