First of all, you are saying TRUSTED_CONNECTION=TRUE. That means Windows authentication. Which means the current user.
Next, even if you take out TRUSTED_CONNECTION, this part UID=domain\ADuser1;PWD=$pass
is not going to work out as intended. When you specify UID+PASS, this always means SQL authentication. That is, the user must be an SQL login. In cannot be Windows user.
When you log on to SQL Server with Windows authentication, you always do that with your current Windows user. If you want to connect as a different Windows user, you first need to perform a login into Windows, for instance with the RUNAS command. (It is also possible that PowerShell provides means for this, but that beyond the scope of the tag SQL Server
.