Stereogenic C and enantiomers
When you say: 

 When I draw the structure of it in such a way so that the 2 CH3 groups are 
 adjacent to each other (on paper ) and take its mirror image I see that I got a 
 mirror image which is not superimposable on the original structure.


and then:

On the other hand when I draw the structure such that the 2 CH3 groups are on        
opposite sides of the central carbon and then take a mirror image of it , 
rotate it by 180° in the plane of the paper I get the original structure.


This tells me you are drawing structures where the bonds are at right angles to each other, like this:

     F                              CH3
     |                              |
 CH3-C-CH3        and           CH3-C-H
     |                              |
     H                              F


This will make the seeing of the structures much harder, you can't quite see the 'tetrahedral'.  In fact both of the drawings you have indicated are the same, only the connectivity matters -- that is the central carbon is bonded to a fluorine, a hydrogen and two methyl groups.

This is a good place to use models.  And if you don't want to buy tinker toys, tooth picks and colored marshmallows (or cheeses) will work just as well.

To resolve your conundrum, expand requirements such that if there exists any arrangement of the groups in your drawing that can form a superimposable mirror image, the molecule is not chiral.

Also consider using dashes and wedges, such as: