Its a continuous tone, but it's getting weaker every day as the batteries run down
No, it's not a continuous tone. My only question is whether it has modulation. I have not read about that, so my bet is that it is not modulated.
You may be thinking of 121.5/243 MHz RF ELT's - a very different beast. They emit a continuous RF signal with frequency swept AM modulation that sounds like an alert siren. Newer ones emit less power than in the past, because the primary beacon is now a periodic digital burst on about 406 MHz, and the VHF signal is intended only for localization as opposed to primary detection.
Burst signals give better battery life. You get better detection with short loud (higher power) bursts than long quieter ones. However, more modern equipment, in theory, can do about the same with either one (by integrating the long quieter signal over time achieving the same SNR as the short louder one).