Should IT department sign a SLA with the other departments to accurate define the scope and level of service?

you should have always SLA's to measure yourself.

However between you and other tems in your comapny this would depend on the level of trust in your company.

if you start doing SLA's between you and other teams openly and communicate these or push back stating" my SLA is xxx" then this means you have reached a level of distrust in you company that will not be bridgable unless a big bang comes like new management or big layoff..etc