This problem seems to be the most discussed problem with new users of tanks. You'll likely get as many different points of advice as you will people willing to give it to you. My favorite way to vape is with tanks. Once you learn how to get them to work, they offer the most trouble-free and satisfying way to enjoy your juices.
I use Smoktech Pro DCT tanks (the polypropylene syringe type plastic that is frosted). I've never liked the clear polycarbonate tanks for multiple reasons. Anyway, these work pretty good for me. The trick, learned from my experience, is to find the best way for the right amount of juice to wick into the carto. Too much juice wicking will cause it to flood and leak, too little wicking and the carto becomes dry and you'll get that burnt taste or dry pulls.
In my experience, a single pre-punched carto (with the tiny drilled hole) is just not large enough to allow enough juice to wick. Sometimes you can find double or triple punched cartos available, which usually do work, but they are hard to find online. Probably the best thing to do is to get youself a good carto punch tool and punch your carto yourself.
The better carto punches hold more of the carto within the tool to prevent the carto from warping from the action of the punch tool. A warped carto will be difficult or impossible to pass thru the tank's endcaps. The Phiniac website
Tools & Accessories - Phiniac Rugged Pyrex Glass Tanks offers this tool for $15, and also offers a handy tank tool that fits on the end of the carto which makes filling the carto and inserting it thru the endcap much easier. Other ecig websites offer carto punch tools, too. Just try to find one similar to this.
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Having your own carto puncher allows you to make the holes the size you want, and choose the number of holes you need. This can depend upon how thick the juice is, how heavy of a vapor you are, and if you vape at higher voltages than 3.7. Sometimes I punch one med sized hole near the bottom of the carto, sometimes two. I've even used three (two near the bottom and another in the middle).
The Smoktech cartos have a flange which will stop the carto in the right spot within the punch tool. You spin the nut clockwise until you hear a little "pop", which means the carto tube has been pierced. Give the nut another twist or two, and then check to see how big the hole is. If too small, carefully line it up over the hole and twist the nut some more. Experience will allow you to know how many times to twist the nut. Don't twist too far or you can damage the air tube in the middle of the carto. Video here shows how easy this is
http://phiniac.com/pid-832/carto-punch.html#video
In this image of one of my tanks, you can see the usual size of hole that I normally use in my cartos. (This tank isn't one of the DCT tanks I have, but of a Tripple V pyrex tank that happens to best show the size hole that I punch.) Click on the pic to enlarge.
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Sounds like you
did try to prime the carto correctly. It's my guess that your hole is too small, not allowing enough juice to wick the carto filler.