Billy Billy Billy...
The strop will do absolutely NOTHING to increase the sharpness of the blade unless you put abrasive pastes on it. In any case you'd want a paddle strop for that purpose (especially as a beginner) and you'd need to get an unpasted strop for its real purpose, which is aligning the edge. Hones actually remove the metal thus making the razor sharper. If you want to cheap out, go to www.redtrader99.com and get the starter set of 3 hones from Tilly.

The strop effect:
No matter how fine the edge is, it still has microserrateons, like a saw or a serrated knife. Since the metal there is fairly thin, they'll be very fine, invisible to the naked eye. After a while they go out of alignment on their own and the teeth are pointing in all directions. Stropping will gently bend them into pointing in the same direction. It will not remove any metal, thus not making the edge any sharper.
The hone:
Various hones have various sizes of abrasive particles. As you move your blade over them, they remove the metal, thus sharpening the edge. The larger the particles, the coarser the hone and vice versa. For your blade you would need at least a medium and a fine finishing hone. If I remember it correctly, the hone you are bidding on is medium grit, but I'm not 100% sure. In any case you'll need a fine finisher like the Lithide (cheapest alternative).
Pasted strops:
These strops are either hanging or paddle strops, with abrasive pastes on them. These pastes actually remove the metal. A paddle strop is easier to work with, so people usually do not put abrasive pastes on hanging strops. They are used for fine finishing and touchups. However if your razor will not shave, it means it needs some work on the hone before taking it to a pasted strop, which you can buy from Tony Miller at www.thewellshavedgentleman.com. They will make your shaves more comfortable.

The reason you are feeling more draw in one direction is that that side of the edge was polished more (honed better) so it is sticking to the leather more.

I hope this info helps.