It's not really a double negative as the negations are in different clauses. If you remove both negations, you get a completely different statement:
> Hiring junior engineers will solve the problem you think you have.
> Hiring junior engineers will solve the problem you think you have.
Refusing to hire junior engineers won't fix your actual problem.
I don’t understand what you mean here.
> Fixed
Your edit is the same in terms of the negatives (though refusing to do something isn’t really the same as just not doing it). But you’ve changed what it says by referring to “your actual problem” instead of “the problem you think you have”.