The freewheel wobble is normal (within reason) and your old one probably wobbled too but you didn't notice. From your description, especially because the lever isn't returning crisply I guess that your problem likely stems from cable friction.

A few possibilities come to mind, especially if you replaced only the inner wire, and continued using housing more than a few years old. Either you need to lube the cable better, or the inner liner of the housing is worn through in the places of highest friction, or the housing is trying to pull through the ends of the ferrules and pinching the ends closed a bit.

There's an easy test to see where cable friction might be the cause. You'll need a repair stand, trainer, ropes strung from pipes,or patient friend to hold the bike up with the wheel off the floor.

Shift the bike to the last gear outside of the problem sprockets and make sure it's trimmed perfectly and pedals smoothly. Now leaving the lever alone, shift one gear by pulling the wire form the frame and using hand touch trim the RD as well as you can, then while still pedaling release the wire and the RD should shift back and trim correctly. Repeat this moving over one gear inward until you've checked each of the middle sprockets both as the high and low choice.

If the RD is working fine this way, but not with lever control, it's either the lever or cable forward of the down tube, if it's sticky or doesn't shift back crisply here, it's either the last loop of housing or something in how the RD itself is set up, possibly the jockey wheel being too close to the middle sprockets among other possibilities.

You can also check the lever in a similar way by parking the RD up on the largest sprocket and without pedaling shift the lever against hand tension on the cable at the downtube, though it helps if you know from experience how it should feel.

Lastly, consider that after riding a well broken-in drivetrain for a long time it's possible that you're oversensitive to the feel of new components that need to be ridden a bit.