Burning smell with check battery and engine light

Our vehicle has had an intermittent problem with a burning smell with no smoke. The battery and brake light comes on at the same time. It seems to happen when it is wet. Like after a big rain fall or recently we had a large snow melt leaving good sized puddles. It does not seem to happen until we drive for a bit. We have noticed an issue with the anti-lock brakes locking at times causing us to pump the brakes. We have also noticed that the headlights seem to be dimmer and the wipers are slower. We had not noticed the smell for quite a while. I think the last time was in the summer, after driving through a large puddle? Then yesterday after a really mild day for January and after driving through some large puddles from snow melt. I have read some other people's issues online and have not found anything that is a definite match. The closest idea I have seen is a issue with the alternator. If anyone has any suggestions what this might be?