While the Talmud does indeed say that, neither the Rambam or the Shulchan Aruch mention it. Chasam Sofer (OC 160) explains why:
The statement of the Talmud (Taanit 29a) "Just as when Av begins we decrease in happiness, so too when Adar begins we increase happiness" is Rav's opinion, following his opinion that the mourning of Tisha B'Av applies to the entire month. Thus, just like by Av the mourning applies to the entire month, so too by Adar, the happiness applies to the entire month.
However, we do not hold that the mourning applies to the entire month1, and therefore there is no longer any reason to say that we should increase in happiness during the entire month of Adar.