Need advice on my 2003 GMC 2500

2003 GMC Sierra 2500, 230k miles.

My grandpa gave me this truck not long before he died several years ago, so it’s sentimental to me. But I can’t get the thing to start reliably for the last 2 years. If it sits without running for a 2-3 days, the battery is dead the next time I go to use it. The battery is only a year old, and the shop insists it’s fine, but they’re unable to figure out why it’s always dead. They just say that there’s a lot of possible reasons it’s doing this and figuring it out could take a very long time, but it’s likely shorting out somewhere due to the rusting underneath and on the body. None of the other vehicles I have are strong enough to jump it, and the trickle charger I’ve been using the last few days isn’t working on it either. Big tow trucks have been and to jump it, but that’s not a reasonable solution to a long term problem.

I’d like to keep this truck, but not if I can’t ever drive it. I hear that this particular model engine is valued by gear-heads, so I don’t just want to bring it to the scrapyard, but I also don’t know where best to advertise its sale, if I decide to sell it, so someone who knows its worth would actually see it.

Has anyone had a similar issue with keeping a battery charged and found a solution?

If I can’t find a solution and decide to sell it, where’s a good place to list it to find someone who would be interested in it?

Thanks in advance.