Instead of pressing down on the nose to close the airway, try to just cover the nostril with the finger and then thumb. Doing this gently might minimize closer.
I also concur with SohamYogaStudio, do Jala Neti - it definitely helps. It’s weird at first but after a few times the nose comes to love it.
As to your question, is it obsolutely neccessary - no. It is just another thing that really helps us on our Path. It helps stabilize the mind, purify the system, balances, increases, and gets the energy flowing, just being a few of the benefits. In the end though, Yoga is not dependent on pranayama.
Also, pranayama is not just about alternate nostril breathing. Check out Donna Farhi’s and Thich Nhat Hanh’s books on pranayama. Both introduce us to working with the breath without all the bells and whistles.
Hope this helps.
Be Blessed…