Why my circuit only draws max. of 2.8V across the electromagnet? (Circuit picture included)
You don't.  Not with those parts.

The voltage across the coil would be around 12V, if you connected to it to a 12V power supply capable of delivering the current that Ohm's law (E=IR) says will flow through the DC resistance of the coil at 12V.  Assuming (for example) that the coil has a resistance of 1 ohm, your power supply would have to supply 12V at 12A.

You need to measure the DC resistance of the coil, and find out how much current your power supply can deliver.

You probably need a bigger power supply to get a higher voltage across the coil.

Then, you will need a bigger transistor.  The 2N2222 can't handle much current.  At 12A, a 2N2222 would pretty much dissolve into smoke.



The thing about coils is, that the strength depends on the current.  You use ampere turns to compare coils.

A coil with 10 turns at 1A is as powerful as another coil with 100 turns at 0.1A.

You need to look at getting more current through the coil rather than at the voltage across the coil.