I have a Mac. OS 10.5.8 (64-bit), Safari 5.0.6, Flash Player version 10.3.183.90

Flash Player 10.3.183.90 (Mac only)  is the latest Flash Player that will work with OS 10.5.8. The reason is that Leopard (10.5) was the LAST "Universal Binary" OS build Apple released (it runs on PPC and Intel Macs). Because of that, the Flash Player plugin is also a "Universal Binary" but since even Apple doesn't support U/B software as of September 2009, Adobe stopped development of the U/B Flash Player in November of 2010.

That's the bad news.

The good news (if your Mac is Intel and NOT PPC) is that you can upgrade to Snow Leopard (OS 10.6) through Apple for $19.99

That will allow you to install and upgrade a LOT of software that won't run on PPC Macs, including Flash Player 11.8