Any Hope for a Wool Sweater I Am Allergic Too?

You might try hand washing it in dawn dishwashing detergent. This is often recommended for knitters when washing and blocking sweaters. What you will want to do is fill a sink or large tub with cold water and the a capful of dawn liquid. Don't agitate/scrunch/handle the sweater more than needed to get it in the tub and wet. Soak it and swish it a bit. Let the water out and rinse it a few times. You can try this in the washing machine but you can't let the washer agitate and be VERY careful taking it out, supporting the sweater so it doesn't stretch. Then lay the sweater on towels if you don't have a sweater rack. Shape it to the size it should be. What I'm hoping is the dawn will take out some of the lanolin or dye that you are allergic to. Your reaction sounds like the problems I have with some dyes. My sister gets hives from the lanolin in wool.

ETA: I have knitted with wool for 35 years. WOOL CAN BE WASHED! The shrinking comes when heat and agitation/friction is used.
http://www.marthastewart.com/269927/how-to-hand-wash-a-sw...

Not all wool is the same. Some wool tends to cause more issues than others. Your first step should be to soak the sweater in cold water with something like woolite. Make sure you rinse it well and lay it flat to dry. This will get rid of any chemical treatment or things that might have attached to the sweater during the manufacturing or shipment of the sweater.

You may be allergic to chemicals used to protect the garment in the store/shipping. Get a wool detergent and wash it in cold water according to the instructions.
If that does not help I don't think there is anything you can do.
Good luck

If the sweater was making you break out, then you might be able to wear a turtleneck underneath it. However, you say that the problem is that it makes your nose run. I don't know anything that would cure that... Have you tried washing it in something like Woolite?

Otherwise, I would put this one in the same category as 4" high heels...looks really nice, but makes you miserable. I would just give the sweater away and call it a lesson learned.

Wool makes me itchy, so I wear a lightweight long-sleeved tee under my wool sweaters.

a long sleeved cotton shirt underneath always works for me, but if your nose is running, then you are severly allergic......to the point of scent, not only touch. So, if you are not willing to take allergy meds, I think you can kiss the sweater good bye. It is clear you cannot wear it. Wool cannot be "washed" it will shrink immensely. Perhaps try dry cleaning it.