Wow. This must be very frustrating! Sorry to hear (read) about your pain!
Lots of great advice above. Just some thoughts:
1) Common problem #1 - juice clogs up atty over time. User doesn't clean out the atties as well as he/she thinks. Atties don't perform well, performance drops off. User "over-vapes" them, they get hot, melt the carts.
You can tell this is it, because when you open a new atty, blow out the primer, put juice in, it works well at the start. Performance drops off gradually due to clogging. Getting it too hot can melt/break the wire too. Multi-meter shows there is a connection if clogged, none if a break.
2) Crap atties that are just overheating, or too low ohms for you (doubt it with standard 510 batteries, but who knows..... Try different atties as mentioned above)
3) bad battery connection - probably not it either, since you are melting carts.
QUESTION: how does the atty fail? All at once (no heat), or gradual reduced performance? Do you have a multi-meter? Can you test a "busted" one?
QUESTION #2: (asked above too) How are you cleaning them? Do you blow them out twice per day?
4) Some juices clog the heck out of atties... real fast too. Try vaping some unflavored nic juice, or switch juices.
So I'd try in order:
1) Blow out atty 2 times per day
2) Clean atty with water, vodka or ISO (and dry out the alcohol)
3) new juice, new (different type of) atty
4) fresh batteries
Just two cents. Hope it helps.
EDIT: Oh, yeah, uh.... Screw the carts. None of that filler crap works. Try dripping. If you like it, get a drip tip...
Stick with it. Once it works out with the carts/atties/juice, you can upgrade the other hardware. It's worth it to get off tobacco!/stinkies!
....... Nowadays I HAVE to hit them for 10-20 seconds otherwise there is no vapor and very little flavor......
Hmmm.... bad battery or bad atty..... Melting carts indicate a bad/clogged atty, with a working battery. But, also clean off the battery connector and atty connector too.