I suffer with headaches and sinus, the medication I take sometimes make me drowsy, In this state of drowsiness could I read Salaat and Qur’an?
In the name of Allāh, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-raḥmatullāhi wa-barakātuh
We advise you to take the medication after performing the Salaah. However, if that is not possible and the drowsiness does not reach a stage were you loose your senses, such that you cannot distinguish what you are reciting in prayer, then it would be permissible to read Salaah in that situation, although it would be better to read Salaah after the drowsiness disappears.
عن عائشة رضي الله عنها : أن رسول الله صلى الله عليه و سلم قال إذا نعس أحدكم وهو يصلي فليرقد حتى يذهب عنه النوم فإن أحدكم إذا صلى وهو ناعس لا يدري لعله يستغفر فيسب نفسه.1
Nabi Sallalahu Allayhi Wassalam said, “When one of you is tired whilst reading Salaah then he should lie down and take a rest until you no longer drowsy. If one of you would have to read Salaah in the state of tiredness, instead of seeking forgiveness he might curse himself.” 
And Allah Ta‘ala knows best
Mufti Luqman Hansrot