Just because there is corrosion around the battery does not mean the battery is bad or that you are in need of a new one. Corrosion is normal with a traditional automotive battery. The only batteries you won't see it with are sealed batteries or gel/dry cell batteries, anything lead acid you will.
To check the battery a parts store may have a battery load tester, something like this:
It puts a load on the battery and will basically tell you if the battery is bad. You can also use a Hydrometer, if you are able to remove the battery cell covers, and measure the acid in the battery, but it is more invasive and not always as accurate.
Also, getting the alternator rebuilt/rewound will not solve anything. All that will do is give it more headroom. It won't change the amount of amperage it puts out when at the loads you are applying right now, read: it won't fix your dimming issue. What it will do is give you more amperage under more load, large stereo for example.
I guess another problem it may be is corrosion in the voltage regulator pigtail, the 3 wire plug on the alternator, or possibly a broken wire on that same pigtail. The common break wire is the one that goes back the the cars computer and tells it how much to charge.
I still think a good cleaning of parts and you should be fine. Also it would be good to find out what the alternator is charging at when they dim and when they are lit correctly.