Corrected entry: Where did Ash's chainsaw go? He uses it once to get out of the pit, once to dismember bad Ash, and then it disappears. Why didn't he use it during the final fight, when a chainsaw would have been much more effective than a sword, even while not running. (00:39:50)
Correction: In addition to the other answer, it's also entirely possible he simply ran out of gas for it, and didn't want to siphon gas from his car to use it. Either way, a character discarding a weapon isn't a mistake in the slightest.
Trivia: Bafflingly, despite having little sex or profanity, and only minor cartoonish violence, the film was slapped with an NC-17 rating when it was first submitted to the MPAA. Everyone involved with the film was shocked. Turns out, a single 1-second shot of a little black, blood-like goo splashing onto a wall following a decapitation was the reason the MPAA gave the film an NC-17. Once it was cut, the film was reduced to an R.
Suggested correction: While it's true that it initially got a NC-17 rating, it wasn't because of a 1-second shot of goo. It was mainly for the decapitation scene and ostensible gore. Director Sam Raimi trimmed down the decapitation scene, but refused studio pressure to trim the movie down to a PG-13 rating, so most of the people involved in the actual making of the film weren't expecting a PG-13 rating.
The one-second shot was from the decapitation scene you mentioned. It's the shot the bloody goo splashing on the wall after he slices the hag's head off. As for the second point, upon looking around, I'm finding conflicting reports. I've only really seen one or two sites saying Raimi "refused" to trim the movie down, and many more that imply he tried to appease them for a PG-13 and had no reason to believe it would be rated R/NC-17 due to how cartoonish the film is (several of these sites also cite a book as evidence, but I can't find the book online). But given there are conflicting reports, I'll edit out the last bit.
Corrected entry: When the army of the deadites arrives at the castle, Evil Ash says something along the lines of "Bring me forth into that castle." A few moments later, he says "Sally forth!" only not being able to say forth because of his decayed mouth, despite being able to say it only moments before.
Correction: It's debatable whether a chainsaw would be more useful than a sword. Ash could have simply decided against using his chainsaw. Alternatively, he could have used it off-screen, but it simply broke. Also, Ash is commonly depicted to be an idiot, so that could explain it.