That said, the debug versions exist for Flash developers to use to debug their applications. They're actually much slower than the regular versions, because they're keeping track of a lot of extra information to aid in debugging. They're not intended for general use (and definitely not general web browsing).
If you want a consistent working experience on Linux, Google Chrome is probably your best bet. The PPAPI interface abstracts away a lot of the fragmentation between various Linux distributions. Without knowing the distribution and version you're using, I can't really make a decent recommendation, but it sounds like a permission problem.