I have a grizzly 14" bandsaw, and have used it to resaw successfully on MANY occasions. You don't need to replace the saw, or go to a lot of extra expense to accomplish what you are doing. It may surprise you, but I use the sock fence, and a 1/4" 6TPI blade to do so. The blade is a timberwolf blade. I have used this setup to resaw the full 6" capacity (I don't have a riser block, and no plans to get one) The key to doing a successful resaw is to let the blade remove the material. Do not push the work piece through too fast.
Here is the process I use to resaw a board into two work pieces:
1. Flatten both sides of the board. Being parallel at this point is not critical, as that is taken care of at a later step. This can be done with a jointer or hand planes.
2. Make sure blade is properly tightened AND tracking correctly. The tracking is the critical component here. If your blade isn't tracking right, it doesn't matter how wide the blade is, it isn't going to resaw well.
3. Set the fence so the blade is centered on the work piece. Note, you will want the two pieces to be reasonably thicker than the final dimension.
4. Make the cut. USE A PUSH BLOCK. While a tall fence would be nice (one can be easily made), I use a push block on the table to keep the work piece firmly to the fence. Again, this is NOT a race. Take your time and let the blade remove the waste material.
5. Off the band saw, you have two boards that have a flat face. Use a planer to make the rough face smooth and parallel to the flat face, and then bring to your final dimension.
6. Edge joint one side of each board.
7. Cut to width.
8. The work pieces are ready to incorporate into your project
One piece of advice, resawn boards do tend to be a bit more prone to movement as you have likely released some internal stresses. Try to get them into whatever glue-up you are doing for them the same day you resaw them, or take other measures to keep them flat until you are ready to use them.