Yay! Another stuttering dev here on HN! I have a stammer too (stutter here in the US); you're not alone!
I'll bet that code tests in interviews really suck. I know that they suck for me.
I think alot of it is going to depend on interviewing with the right person. Keep trying, and eventually you'll find them. Also, if that doesn't work, you can always try to pick up totally remote freelance work, but if you're looking for full-time, you should be able to find it.
One thing that may really help is getting an introduction from a friend (or a recruiter) to any companies you're interviewing for. I got one job with a conversation that went like this:
My friend: "Well, I have someone for you, but he has a pretty bad stutter"
Employer: "As long as he's smart and can get the job done, I don't care if he talks backwards!"
And I got the job.
Real employers (who aren't skeezy and exploitative like your current ones) don't care how you talk, they just want someone who can get the job done. And those are the people you should be working for.
In some ways, you can think of the above as a positive selection filter. Imagine all the non-stuttering schmucks who are out there working for skeezy employers who care more about how people talk than who can get the job done (and imagine how shitty the management must be if their hiring is not based on people's skills).
In closing, I know this turned into kind of a rant without any really good advice, but I just wanted to give a solidarity shout out to another stuttering dev here on HN.
Now, if you'll excuse me, I'm going to drive-by down vote all the commenters who are like "herp derp why don't you just stop stammering"