Use dipole antenna in monopole design
A dipole antenna requires a balanced feed for it's impedance and emission shape to remain good. A monopole is fed with an unbalanced signal i.e. signal and (local) ground) - it's basically half a dipole.

People use what is called BALUN to convert an unbalanced signal to a balanced signal and, by the sound of it this is all you need. It can be very simple - a couple of turns of the coax feed to the antenna wound around a ferrite toroid might just do the job or a little RF transformer (can be easily hand wound) will also suffice.


  If I build the dipole antenna myself from two pieces of wire, do both
  halves of the antenna need to be aligned directly in one line or can I
  place the halves apart from each other to where they are convenient on
  the board?


There are many variations on a the theme of dipoles and there are such devices called folded dipoles that bend the two limbs around in a circle but, it's always best to keep both limbs as symmetrical as possible and far away (as much as is feasible) from other circuitry.