Of course a lot of it was propaganda, they needed to exploit what they could to get common support for the war, however:

-Litvyak started on Po-2's, trained on them, then instructed others in them, finally making her way to Raskova's regiments.
-After initial training with the 586 IAP, she was transferred with other pilots, like Yekaterina Budanova, to different regiments along the front lines.
-The 586 IAP was a Yakovlev squadron, and they flew Yaks from the beginning of their careers to the end of the war. Litvyak was transferred to a squadron that flew the La-5 for a brief time (take note that the LaGG and La series are different planes) After being in a few different air regiments, Litvyak transferred to the 296 IAP. Here she flew more Yaks, the Yak-1 and Yak-1b. She DID fly Yakovlev aircraft for her duration of the war. As far as I know I have not found anything stating she flew any other aircraft in combat other than the La-5 and Yaks.
-The official word until Gorbachev came to power was that Litvyak, whose body was not yet found, had become a prisoner-of-war and therefore, an undesirable traitor. When they later found her body she was given a place of honor.

The women who flew the Pe-2 planes were part of the 125th Guards Bomber Regiment. The regiment was actually not all female, and a lot of men served in the crews and regiment. In fact, the CO was male.

The women who flew the Po-2/U-2VS were part of the 46th Taman Guards Night Bomber Regiment. This was the only all-female regiment of the group of three.