Am I sinful for using the internet in the institution for disobedience, knowing that I was the one who proposed connecting the service and signed the contract in my capacity as an engineer, and what should I do to clear my conscience?

If the company permits managers to use a certain service, and they do not use it for forbidden purposes or waste work time, then there is no blame upon you. As for what some employees do that is contrary to this, you must warn and forbid them. If they do not respond, then you must inform the management. This falls under enjoining good and forbidding evil. By doing so, you clear yourself of responsibility.