For those curious, I think I have figured out the effects of a low battery and not changing it out in a timely manner and what sensors it will eventually affect. I have had a low battery warning for about 6-7 weeks now. I have just been too lazy to go up on the roof and switch it out. I will be doing that today. Around 5:00 this morning, all sensors crapped out until the sun came up. Last week, it rained about a tenth of an inch, but nothing registered. So, I think it goes as follows in this order: rain bucket, followed by temp-dew point-humidity, and lastly all aspects of wind measurement (about 10 minutes after temp-dew point-humidity stopped recording).
Long story short, just change out the battery as soon as the console alerts to do so.