If you CRT is dying fast, and the options to connect your Amiga to a modern monitor are not easy to come around, you should take a look at the A520HD, the HDTV equivalent of the classic Commodore A520 TV modulator. Instead of encoding the Amiga video signal to a low-quality composite output it converts the RGB output to a high-quality YPbPr signal compatible with HDTVs with a component input. The A520HD connects to the Amiga 23-pin video port and is powered directly from the system.
GGLabs produces open source hardware so all the documentation of this adapter is available at their website, allowing you to build your own. If you, like me, has no skills to do so you can buy an assembled one for US$70 (click Shop at GGLabs website). You can also find GGLABS products on ebay.