I also find rechargeables work rather poorly in my weather radios. Like you, just a matter of days, plus the radio dies all of a sudden. But with alkalines, mine will go maybe 10-14 days, and for the last couple days, the low battery light flashes.

This particular application is one situation where alkalines have an advantage over NiMH in my opinion--the gradual decline in voltage is long so you have a chance to catch the low voltage condition before the battery completely dies. With the NiMH batteries, I'll go to sleep with the light blinking green and wake up the next morning with the radio dead. Never had a chance to see it go into that low battery mode.

I have read that the newer low self discharge (LSD) NiMH, like the Sanyo Eneloops should do better, but I haven't gotten around to buying any yet.