It certainly doesn't sound like your iPod has any water damage. I've never heard of this happening from sweat before. I assume they're just making up excuses so they can claim that you are no longer covered under the warranty.
At any rate it sounds like either a battery or a hard drive issue. I would be happy to take a look at it for you. I run an iPod repair business. My company website is: www.fargotripod.com
If you are interested in sending it in just fill out the repair form and we'll send you all the information you need along with a free quote on the cost of repairs. It sounds like we should be able to get your iPod working like new again in no time at all. Let me know if you have any questions.