Do the antihistamine nose sprays and the steroid sprays lose their effectiveness over time? Should I alternate, using one type of spray each month?

Should not. These sprays should not lose their effectiveness over time. Some people need both in combination. I would suggest using the antihistamine spray and adding in the steroid spray when your symptoms are not controlled. It may take a couple of days to feel the full effect of the steroid spray.
Nasal sprays. Antihistamine & steroid nasal sprays tend to maintain their effectiveness over time. Decongestant sprays, like their oral counterparts, however, develop rebound, some more quickly than others. Rebound is the effect whereby repeated use actually causes the opposite effect—congestion. Consult with your pcp or allergist about nasal spray use. Perhaps a combo of oral med w sprays will be optimal.