All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
If the situation is as you mentioned in the question that you performed a complete Ghusl, then you should not doubt after this about water not reaching the roots of your beard, especially that you started doubting after you finished your Ghusl. The scholars stated that doubting after having finished an act of worship does not have any effect. The same thing (doubting not having any effect) applies to when one has so many doubts. Besides, doubting after Ghusl should not be taken into account. Therefore, your prayer is correct and you are not required to do anything.
For more benefit, please refer to Fataawa 95488, 92050 and 83901.
Allaah Knows best.