It is the fungi that decompose wood that need the nitrogen. If they can get it from the air, they will. So as long as your wood chips are not buried underground, the nitrogen in the soil should remain unchanged. I would continue to pile on the chips.

Look up 'ramial chipped wood' on the Internet.

I've grown tomatoes/peppers/etc in woodchips for years, i wouldn't worry too much. Add grass clippings or some other source of N if you need to.

Wood chips are o.k. For really steep slopes I reccommend erosion control blankets. Lots of well-rooted, shade-tolerant plants will help with the long-term solution. I used an erosion control blanket and creeping red fescue to rescue one steep slope. If your slope is really steep, you might have to do something more drastic like terracing. A native legume could help add nitrogen. You can also check out the hillside forum to inquire of other hill-minded people.

If you can find coarse (coarser the better) hardwood mulch it will interlock and resist floating or washing away.

It simply isn't true that wood takes soil nitrogen, notice I say soil nitrogen; it may in fact add nitrogen. Fungi is the main wood composting organism which is eaten by other organisms that take nitrogen from the air then these organisms are eaten releasing the nitrogen taken from the air further down in the soil. As it works out this nitrogen is readily available to the plants while chemical nitrogen may simply wash away with little value.

Seeding (your choice of plants) on top of spun bonded fabric will hold a slope very well. I've seeded on top of cheap (which will allow roots to go through) fabric that causes the roots to go through the fabric holding the whole thing in place. Lay the fabric top to bottom instead of cross ways. This may require mulch of some type on top of the fabric to help seed germination until it is established.

If the base soil was very low in organic matter levels and a lot of organic material was placed on top of that soil the bacteria and fungi that should be working in the soil will see that food source and go after it, taking with then any Nitrogen (protein for them) that might have been available to plants. If the soil had sufficient levels of organic matter before the wood chips were placed on that soil you would not have seen any change in plant growth.
So what you need to do is provide the Soil food Web with the food sources they need to do the job they do until your soil is balanced and they don't need those supplements. Whether adding a "complete" fertilizer would help will depend on what the soil under the wood chips is and that soils ability to hold what is essentially a water soluble food source. Some form of Nitrogen might help get the wood chips digested faster but may not do anything for the plants.
What did a good, reliable soil test say about that soil?