I have just recently flattened out a textured ceiling. Was easier than replacement.
Knock off as much of the texture as possible (dust mask on) sand it down, give it a good wipe to remove all dust then apply some premixed drywall compound over the top until it smoothes out. sand then paint.
Redoing drywall is also easy but about $8-$20 per sheet plus compound.
What I would suggest is, get some compound and try it on a small test patch and see what happens. If it comes out nice and smooth carry on
