The paperwork you signed should have made it clear who was responsible for the cost of the insurance.
Normally there is a requirement to make the first payment within 45 days. That requirement was loosened due to the COVID emergency.
Ignoring the emergency changes, the system doesn't require the payment to come from the former employee. COBRA payments can also be made by the former employer, the new employer, a union, or some other third party.
The paperwork should have made it clear how the payment was to be made, including how you would be billed if you had a responsibility for the payments.
Speaking of payments, the paperwork should have made clear what it would cost you. There is no rule of thumb. I have known companies that normally pay 100% of the premium for their employees; so if they went under COBRA they would now have to to go from paying 0% to paying 100%. Other companies pay 100% for the employee but a lower percentage for the rest of the family.
The paperwork should have also given you the method for canceling the coverage.
Please review all the paperwork you submitted, and any other documents you received related to this situation.