Used in the past to show deep respect, now it may sound out of place or overly formal. Your aid has been invaluable - Your obedient servant, A. Hamilton.
Pro Tip 2/3
Not Literally
You're not claiming to be an actual servant; it's just a respectful sign-off. Thank you for your guidance. Your obedient servant, Meredith.
Pro Tip 3/3
Formal Context
Best used in formal letters, especially when addressing someone of higher status or authority. I await your instructions. Your obedient servant, John Smith.