I'm allergic to it too.

Buy some clear nail polish and coat the jewellery in it, it should stop it from reacting on your skin.

Cheap metal is coated with rhodium to keep it from oxidizing, that may be what you're reacting to. Unfortunately just about all commercial metal is treated with it, including jewelry. 

You may be able to paint over small parts of exposed metal with clear nail polish. Or....get plastic snaps and zippers etc and have someone swap the out for you.

I would tell you to look on Etsy, personally. The sellers there are not huge manufacturing corps churning out products as fast and as cheap as possible; they are individual craftsman who care more about their customer being happy than anything else! People on Etsy tend to be more considerate to allergies and requests, so I would advise you to look there.

Aluminum findings and 100% aluminum jewelry is gaining popularity heavily. It is ultra shiny like silver but has absolutely no ability to tarnish, is lightweight, so very cheap, but extremely sturdy. There is a plethora of aluminum-based jewelry - whether in part or whole - on Etsy as well as some stainless steel. You may find someone who is willing to do a custom piece for you with aluminum or steel findings there. It might cost you extra, but it will be worth not having the allergy to it!

The best way to deal with any allergy is to take matters into your own hands. For food, it would be preparing your own food so that you know exactly what it has come into contact with and what is inside of it. With metals, it is sometimes making your own jewelry or even taking jewelry that you've bought and replace the parts you're allergic to such as clasps or chains.

Do some research. First figure out which types of metal you're not allergic to. Then you have the choice to just punch it into google "Stainless steel scene bracelet -fabric" or perform targeted searches on specific websites.

I will warn you that unless you find a website that caters in high quality materials with the style you're looking for, you will still mostly find garbage jewelry. Which is why I recommended making your own. The people (mostly teens) who wear the style you're looking for generally settle for cheap and poorly made accessories. Hot topic is a very prime example of that and the success behind such a business model.  The kind of metal you likely need will inevitably be much more expensive than silver/gold/black plated nickel as well.

I myself cannot wear cheap jewelry due to my own metal allergies. Thusly, I have more expensive pieces and fewer of them. I will also take a very expensive pendant (such as my mother of pearl pendant surrounded in quality metalworking) and place it onto a plain black chord. It saves a bit of cash and doesn't look terrible with the right chord. 

From what I understand, the metal most often causing allergic reactions is nickel. It's not really a question of "nice" or "not nice" metals being used, as nickel can do all kinds of wonderful things when alloyed with other metals. In jewelry it's typically used for corrosion resistance.

Since it's so very common, what you may need to do is find a supplier of nickel-free hypoallergenic hardware and replace snaps in all your accessories yourself. That way you can still buy whatever you want for looks while not having to suffer the rashes, and for just a little more work.

You could also remove the snaps altogether and use leather laces or some such to fasten them instead.

It'd be nice to know what metals you are actually allergic to.