under $35 is going to limit your options...
you can get a cheap set of diamond stones shipped to you for $15 from ebay... this will remove metal and can restore a damaged edge.... it should leave a better edge then what the accusharp is leaving you.
If you haven't done free hand sharpening before, you'll either want to get a DMT guide ($10 or so) or make a paper wedge (fold corner of paper in half, that's 45... fold that in half... that's 22.5... or close enough for you for now) to put under the blade to see what it should look like. Practice and watch some youtube videos.
Your at $25.. you need some ceramic for a more refined edge, or you can improvise a strop with some mothers mag aluminum polish (~$5... walmart by the automotive section...) applied to some cardboard from a cereal box. Cut a long strip from the box, decently wide, and rub the polish in and let it try a little...
Okay, you are around $30...
Now, if you don't damage your blades (they are just dull), and you are trying to keep them sharp, you can get two of the hunter honers or the block sharpeners. Both have videos on youtube. You'll have a sharper edge than the accusharp and it's less damaging by far then that accusharp. Your knives will last longer, but you won't be getting better with your sharpening skills.