Covalent bonding is a complicated thing. The laws governing how electrons behave in atoms and molecules are the laws of quantum mechanics, and so an attempt to understand these things in terms of 'orbiting electrons' is likely to fail. The reality is that electrons don't have well-defined positions and velocities --- you simply cannot say that your electrons are here at one instant of time and move round to here at a later time.
Instead, the most complete description of our electrons we can give is not their postions and velocities but instead an object called their wavefunction. A wavefunction can be thought of as a probability density function for the position of a particle. That is to say, it is a function (just like $x^3 + 1$ or $2^x$) that describes the relative likelihoods of finding the electron at each point in space.
For an electron involved in a covalent bond, its wavefunction might look something like this
where the ticks mark the positions of the nuclei, the horizontal axis corresponds to position, and the vertical axis corresponds to the value of the wavefunction --- i.e., to the probability of finding the electron at that position. As you can see, the electron is most likely to be found somewhere between the nuclei, holding the molecule together. If you like, you can imagine that the electron 'spends most of its time' in between the two nuclei. As I said, the electron doesn't have a well-defined position or velocity. But if you must visualise it classically, imagine that the electron is jittering about between the nuclei, rather than orbiting anything. Electrons don't orbit, and it's silly in my opinion to teach people that they do.