I personally haven't used Touchdown, but I have a couple techs in my department who swear by it.
I'm basically 24/7 (sys admin) so all my email is push. I've honestly never even looked at how long you could set sync times for, so I was surprised that the longest most apps will let you go is one hour. That's just crazy.
MailWise has a "never" sync option that might work for you. It also has a bunch of options available when you swipe like delete, move, star. Nothing programmable unfortunately, but still better than just delete.
Galaxy S4 Mini - SGH-i257m on AT&T LTE
Nexus 5 32 GB Running Kangakat