I was on Baclafen for a number of years but switched to Zanaflex. The main side effect I noticed from it was a weakening feeling in my legs. For someone with MS, that is extremely disturbing and genuinely effects daily life.
It leaves the system between doses, so keeping your schedule with it will ensure the most consistent results for you. I took it three times per day at 10mg per.
It didn't cause me a marked sense of drowsiness, but remember, MSrs struggle with fatigue all the time anyway, but I can't say that mine was worse during the Baclafen years. I was at 30mg per day at my max.
I hope this helps a little, and that it works for you!