My hunch is a problem with hot line vs switched hot load. On the old switch it didn't matter. Disconnect all line and load. Separate them carefully and turn breaker on and figure out (non contact tester) which one is hot. That goes to line along with all other hot/switched hot wires except for the one that is for the actual switched load, which goes to load of course.
Looking at the picture, it appears that Line and Load are "two wires next to each other" and not specifically labeled/designated wires. The implication is that the switch grabs power wherever it can, presumably by monitoring both wires since one of them will be hot, and when the switch is turned on, it creates a circuit between them. Which means that there is no one "right" configuration, unlike many other switches where a particular wire or screw must be connected to hot and a different wire or screw must be connected to switched hot. Actually most such switches will use a different color (typically red or blue) for the Load wire to make it clear, so seeing black for both wires confirms that Line and Load are "whatever".
From a logical perspective, there are only a few combinations of 3 wires and that limits what must be going on here (unless something is just plain wacky):
A & B are the two black wires from the switch. Doesn't matter which is which, so considering A as "connected to one other wire" and B as "connected to two other wires".
- A = incoming hot, B = outgoing hot + switched hot - I believe that is what you have based on behavior of the switch)
- A = switched hot, B = incoming hot + outgoing hot - that is what you want
- A = outgoing hot, B = incoming hot + switched hot - that would have the originally switched receptacle always on and the other receptacle (that wasn't switched before) now responds to the switch
So by process of elimination, you must have the first configuration. So confirm that "A" is always hot, and assuming that is the case, move one of the "B" wires to connect with it. That should either work as desired or result in only the wrong receptacle switched, in which case swap it for the other one.