Need help with toilet not draining

You have a 3-year old (deep sigh) - odds are pretty good it's not a 2-week old subscription card (ought to be pretty much mush by now) but some other object likely flushed more recently.

My slow-toilet (unreliable) magic trick is dishwashing liquid, which sometimes (but not always) lubricates things enough to make plunging far more effective. Just a little is all it takes, and using vast amounts won't help more. A cup of bleach overnight sometimes also helps, presumably from breaking things down a little and freeing some sludge. Verify that you can (or refresh your memory on how to) unlock the bathroom door from outside, and lock it while the bleach is in to keep the 3-year old out.

Pouring in hot water can also help, but overdoing that could damage the wax seal, and then you'd be pulling the toilet - which is one more option for desperate cases, albeit one most people prefer to avoid. But if there's something like a doll's head or golf-ball down there, a snake may not grab it, and then pulling the toilet is the only thing that will work. I would agree with @Keith Hoffman that it's nothing to fear, and the tools required are far more general-purpose than a snake (wrenches, mostly.) It just crosses into the "icky" factor that makes plumbers so much money ;-) - face that and save. Sponges and rags are an alternate way to get the water out, and remember to have a plastic bag stuffed with something (other bags or paper work) to plug the hole while the toilet is off (and don't forget to take that out before replacing the toilet.)

The main reason (AFAIK) to strictly avoid Drano® (et al) in a toilet is that you have a MUCH higher chance of causing yourself (or the 3- year old if she gets in there while it's in) serious chemical burns from splashing in a relatively open toilet, rather than a more contained sink drain. My personal experience from sinks is that it usually doesn't help anyway, but does make the next plunging session much more worrisome.

Even though you snaked the line near the toilet, it sounds like the line is still blocked or slow draining such that standing water sits in the line above the point of connection to the vent. Vent lines for toilets are often connected as shown:

Standing water above the reducing wye where the vent connects will cause water to be pulled from the toilet bowl as water flows down the line (such as when the other connected toilet is flushed). A flood when you flush both toilets indicates the sewer line is slow draining and cannot handle that much flow. Possibly the vent is plugged. More likely there is an obstruction such as tree roots infiltrating the line. If you have a septic system then it is time to pull the inspection lids, especially the one at the inlet. Check for obstructions such soap caked near the intake baffle.