(Mis)understanding on MOT

In VBT, one can understand the example of hydrogen like this - Two hydrogen atoms come closer to each other, and their orbitals overlap, stabilize, and ultimately result in a bond. This is not too difficult to understand, and can easily be visualized and seen in an intuitive manner. 


MOT however, is different. Instead of the two orbitals overlapping each other to stabilize the molecule, they both mix together to form a bonding and anti bonding orbitals. Alright, the electrons from both the Hydrogen atoms go into the lowest energy orbital, which is the bonding orbital. 


My main doubt however, comes from the electrons of both atoms being SHARED in a covalent bond. When two orbitals overlap in the same phase, they add up to each other (BMO). And when they overlap in opposite phase, they cancel each other (AMBO). So, two hydrogen atoms overlap in the same phase, and they make a bond and so stabilize each other and form a bonding molecular orbital. Wouldn't this mean that both the electrons are shared between the two atoms and hence both their orbitals are filled? Is this a false understanding of how bonds are formed?


Does this have to do with the mathematical equations themselves? They're as of now, too advanced for my course. 


