A few suggestions:
If you have records of your past rental payments, submit those with your application to show that you did indeed pay rent on time. Even if you can't get a response, from your previous landlord, these records make you more attractive as a tenant.
Have a boss, manager, or supervisor (not friend or family member) write you a letter of recommendation as to your character. It shows that you're employed and you could ask that they highlight such qualities as responsibility, reliablity, etc.
Contact your previous landlord and keep contacting them and contacting them and contacting them until they respond to applications with rental verification.
Sublease - then ask if you could use them for your "new" previous landlord and have them verify rental history.
And finally, this isn't a suggestion, but a comment on the housing situation here.
It is HARD to find a place. On average? I'd say it takes about 4 months. That's how long it took me, but is similar to what I've heard from friends, acquaintances, and reading about other's experiences on here, Nextdoor, etc. My current roommate found me after a 6 month search. Even treating the housing hunt as a full time job, it is difficult to find availability and difficult to make it as the first in line. It is discouraging and disheartening. But it is possible.
I hope something turns up for you and best of luck.