You should (typically) hear water gurgling in your sink's drain when the DD is draining. Or you could disconnect the drain hose from where it attaches under your sink and run it into a bucket, if you feel up to giving that a try. It's possible the water was up to the vent, there's a deflector mounted behind it to help prevent water from spilling out ... although it still would overflow at some point and/or from sloshing when the drawer moved. Once it overflows, the machine's flood-sensor (in bottom of the cabinet) would trigger. Open the upper drawer, there may be a model/serial sticker at the left just above the lower drawer.

A small blower (as you probably know) runs during the dry period to pull room air through the DD and exhaust the residual heat/moisture. The vent is at front/center beneath the DD. Depending on where/how your storage drawer is positioned in relation to your DD's exhaust, you're probably having some warm/humid air seep into the drawer. Leaving the storage drawer ajar as you've been doing allows it to dissipate better. Doubtful there's anything wrong with your DD. Probably the only workaround is to cancel the DD cycle when the drying phase starts and pull it fully open for flash-drying instead of the normal airflow drying. Note that if you do that, the DD's pump will likely run in drain mode for a minute or so when the DD is next closed.

A 24” framed drawer stack is too small in width to be a good dish storage cabinet. The interior space of the drawer will be around 18”-19”, depending on hardware choice. In frameless, you get more like 21”, depending on drawer material choice. The most used dish in any kitchen is the dinner plate, at 11” round. That leaves right at 6”- 10” for either bread plates, or glasses, or cups. If glasses and cups is your choice, as they are actually used more than the plates, then the 24” drawer is significantly undersized. You don’t stack glasses and cups. That means having a much wider heavy duty drawer to combine both those and plates, or another drawer, with glasses and cups in the top drawer. That’s at least a 10” drawer, and it won’t align with the standard 6” top drawers of most cabinets, breaking the line around the kitchen. There are ways of doing this, but there are going to be trade offs on function vs aesthetics.

You have two single-drawer units? Or one double-drawer unit? "Over 10 years old" covers a lot of ground for the various model revisions. Citing the specific model number(s) is more helpful than an imprecise age reference. Assuming you don't have one of the very early models that have inflatable seals ... The seal (for each drawer) is part of the drawer lid and does not contact the drawer rim until a cycle is started and the lid motors crank the lid down. The lid attaches to lid yokes at the left and right of each drawer, which in turn are connected to the screw-drive motors to raise and lower it. Could be one (or both) of the yokes on both drawer have broken. Could be one or both of the lid motors on both drawers have failed. A drawer can be forcibly opened if the lid motors fail and the lid doesn't open ... but it cannot be closed again without damaging the seal ... the lid motors must first be manually cranked to raise the lid. Could be the link support arm (which supports the wiring harness, fill hose, and drain hose) has broken on both drawers and something is dragging on the drawer ... but you didn't cite the model to confirm if you have the old-style plastic link support (which will eventually break) or the more-sturdy revised metal support. There may be a few other possible causes.