In Europe, these problems still exist, certainly, but, it's a lot easier to find places/companies that can be deemed as healthy, or let's put it like this, that aren't making the news headlines.
I would propose that you do your due diligence and investigate the business, company, maybe even some of the C-Level people... It can be perfect or terrible, it just depends on each and every individual case.
About the offers themselves: Double the current salary feels a bit too good to be true, so, if a large part of this salary is paid out in equity/stocks, this can tie to the point above about evaluating the health/runaway of the company and how it's been doing over the last few months at least.