I would keep it. Why should anyone else have it? If it just magically appeared at the end of my bed overnight, then fate dictates that it was meant for me. And if not, then fate would dictate otherwise, and it would disappear just as quickly as it appeared.
I'd be tempted to read it, if it meant a wealth of knowledge of events past, present and future. I realise of course the Elder Scrolls are like Tamriel's 'Pandora's Box'. The problem is if you tried to tell/warn people about said events; I imagine in the real world people would perceive you the same way we perceive the NPCs in-game who read the 'Black Books' - nonsensical madmen.
Still - i'd have a thing that nobody else has. Why get rid? Unless I could sell it for millions and millions of pounds. Cosmic, transcendent knowledge is well and good, but in this world, money's more useful.
I'm hesitant to even begin to assert what I could possibly do with an Elder Scroll if I found one. Feldir the Old managed to banish Alduin which is pretty epic, the Grey Fox was able to use one to lift Nocturnal's curse on the Grey Cowl, again pretty epic and useful. However, all the Dovahkiin managed was to find the location of a cave which by comparison is less than spectacular.
Maybe with my luck I'd be able to foresee what I will have for dinner. Wow! Thanks Elder Scroll. The lottery numbers would have been more helpful.
Maybe it was intentional? Perhaps Feldir was secretly pleased his heroism and Tusken Raider-like scroll wielding would be visible for eternity? "YOU ARE BANISHED!"
I didn't want to count that effect the Dovahkiin managed to produce because it required another component. While I may have an Elder Scroll, the original post made no mention of a time wound.
First..I would panic!
After that I would wonder If I haven't been playing too much (Have any of you ever walked past a field of flowers suddenly imagining yourself picking them up as alchemical ingredients? I know I have)
Then I would poke it with a stick or something smart like that , then go to Tamriel and find out who's trolling me I guess.