Yes muscle relaxers do cause drowsiness and can me you tired even the next day.
So make sure when you take muscle relaxers to only do so when you do not need to drive or do anything that requires your full attention.
You can be drowsy and tired the next day with the muscle relaxers as it can take some time for the muscle relaxers effects to wear off the next day.
So if you need to drive the next day early in the morning then do not take the muscle relaxers or at least have someone available to drive you such as a bus, Taxi, Uber etc.
When I started taking muscle relaxers I was so drowsy the next day and eventually my body got somewhat used to taking the muscle relaxes but still I was groggy and drowsy on most days.
So I got my doctor to take me off the muscle relaxers and put me on something different.