Try using this formula:

=IF(OR(AND([Mil Status]="AGR",[Composite Score]>=90),AND([Mil Status]="DSG",[Composite Score]>=75)),TEXT([Previous Test Date]+365, "MMM YYYY"),(IF(OR(AND([Mil Status]="AGR",[Composite Score]>=75),AND([Mil Status]="AGR",[Composite Score]<=89)),TEXT([Previous Test Date]+182, "MMM YYYY"))))

Also, you can add months & years in date as well. Check below documentation for more information & syntax:

Calculated Field Formulas

Try using this formula:

=IF(OR(AND([Mil Status]="AGR",[Composite Score]>=90),AND([Mil Status]="DSG",[Composite Score]>=75)),TEXT([Previous Test Date]+365, "MMM YYYY"),(IF(OR(AND([Mil Status]="AGR",[Composite Score]>=75),AND([Mil Status]="AGR",[Composite Score]<=89)),TEXT([Previous Test Date]+182, "MMM YYYY"))))

Also, you can add months & years in date as well.

Example: Adding 3 months to [Previous Test Date].

=DATE(YEAR([Previous Test Date]),MONTH([Previous Test Date])+3,DAY([Previous Test Date]))

Check below documentation for more information & syntax:

Calculated Field Formulas