?Sizzix Red Die Machine?

I used to have this machine until the handle broke.  You can cut the thin dies.  You just need to make sure that the magnetic adapter/converter piece is included with the machine to cut the thin dies and use embossing folders.  You will probably need to use paper shims or I had a piece of something like plexiglass that I put over the die (so it wouldn't be cut) if it was thin/had a lot of details.

You would use the Sizzix Adapters or Sizzix Converter. You could use cardboard to rig it, this lady has an Instructional Video (it was linked above too). 

All you need is the "System Converter". It's a block of metal with magnets that mounts to the top plate. If you get an original Red with that, it will cut thin dies just fine. I use one as my main machine, and have a replacement boxed one waiting in the closet should it ever fail.

Personally it's my favourite style of machine for cutting small dies. You don't have to crank a full sandwich through to get it to work, just one good thunk and it's done. I do use a top 'plate' with mine made from retired kitchen cutting board so I can cut dies face up and provide some extra pressure. Since I never cut into the bottom plate I have magnetic sticker sheet on there to hold dies and the only consumable part is easily sourced kitchen cutting board.

Me too! Off it went to Goodwill.

Then I discovered a drawer full of the thick block dies. Still don’t know if they can be used with a different machine or if they are destined for eBay or the goodwill too.