I am in the exact same position.
I never had a lot of friends, but over the last year in university, I had an extremely painful and traumatic episode with my second-best friend, which resulted in me and her not interacting anymore. This had the effect of tearing apart my mental health. In the second year, I just had no time for any clubs and societies at all. Like you, I want to have more friends and, especially, be in a relationship, as I feel incomplete without it. But this is made difficult by severe social anxiety resulting from severe trauma in the past.
To mitigate this, the first thing I did was address the root cause of my problems. This resulted in me being booked for a course of CBT and a suspicion I had C-PTSD.
For the third year, I found two exciting societies I could join. One of them was a large biology society, and the other was a high-intensity sport called lifesaving.
So yes, first identify what is causing your problems. Are you anxious about meeting new people? Do you have social anxiety? I have personally seen the terrible effect this has on me, dozens of times.
Second, carry on and try joining a few new clubs to align with your interests with the intention of meeting new people