I don't think an air purifier is actually going to cause dry air but if the purifier has a high powered fan, the swirling air could make it feel drier... moving air will dehydrate you faster than stagnent air.
I have a honeywell tower air purifier
that says it works for 200 sq feet but I think it purifies more than that. I also try to follow Red_34's advice by Vacuum>Dust>Purify to get the best results. The purifier I use also has an IDF filter that filters out airbourne germs so I try to keep it running all the time. I'm thinking of getting another one to increase the coverage in the house.