[dotnet] Close dev tools session safely

[nvborisenko]

Description

To avoid potential deadlock we need execute async methods via wrapping it as Task.

Motivation and Context

Related to #10008 and #12592

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)

Checklist

• I have read the contributing document.
• My change requires a change to the documentation.
• I have updated the documentation accordingly.
• I have added tests to cover my changes.
• All new and existing tests passed.

I have tested the fix, deadlock disappeared.