Changed the rear brake drums and now the car brakes when I press the pedal down further than before. Any idea?

Entirely flushed maybe, but maybe not entirely bled of AIR. 

Many people who have troubles like this do not have a thorough understanding of how to bleed the hydraulic system. 

How did you do it? 

The other possibility is that you did not adjust the brake shoes properly.  You turn the self-adjuster to expand the shoes and remove the gap between the resting brake shoes and the drum.  You can leave extra space, but then the brake pedal goes lower.  You can NOT make it too little space, because the brake shoes will be in constant contact with the drums, and will heat up and expand and cause the brakes to be even tighter, then they might lock.  It's a rather touchy adjustment.

How did you adjust your brake shoes? 

Also, you installed old drums on new brake shoes.  That wore the center of the friction material on the shoes, because your old drums were worn, and therefore had a larger inside diameter.  Then you put new drums on the oddly worn newish shoes, that then contacted the new drums away from the centers of the friction material. 

Maybe you could read online and get a better understanding how to adjust the brake shoes. 

The brake shoes need to be adjusted to close the huge gap you left after you contracted the brake cylinders.. Watch a you tube of rear brake adjustment  for your car. You got this far, so going back to finish the job right won't be too hard. Don't have the shoes dragging, but the gap needs to be much smaller than what you left.

Before you put the DRUSM back on you must ADJUST them TIGHT with a special tool to take up the SLACK< JACK and they have an ODD adjustment device except for VW drums and many MITSUBISHI drum replace the BRAKE FLUID fully as it is a DEAD FLUID after about 2 years of use! As a matter of fact on  TESLA CARS the only service needed is to replace the BRAKE FLUID every TWO YEARS! BYE NOW! 

I take it when you replaced the rear drums you also replaced the rear brake shoes. Now days no one arcs the brake shoes so that the shoes fit the brake drums perfectly. The outer edges of the brake shoes contact the drums while braking but the inside portion of the brake shoe will not be contacting the brake drum until the outer ends of the brake shoe wear into the drum a bit and the middle portion of the brake shoe starts contacting the brake drum. This is fine but the brake shoes will need readjusting. If you pump the brake pedal and it starts getting solid higher up it confirms you need to readjust the brake shoes.

It's not air, it's simply that the rear shoes need adjusting. At the moment they're too far from the drum surfaces, creating the "lost motion" you describe.  Others have described various ways of doing it.

Drums need to be adjusted.

You took off the drums meaning you know about the star wheel which is the adjuster.  When you slip on the drum, you MUST ADJUST THAT WHEEL BY rotating it while you adjust until you hear it scrape the drum noise.   Now do  the other wheel the same way.   Now, you find a strip of abandoned road and back up about 10-20 MPH and stomp on the brake pedal.  Do that several times as the star wheels are now being adjusted by the self adjusters inside each wheel.  A half dozen times is enough.  Now drive forward. Brake and those are the brakes you got.  The rear wheels never should lock up into a skid...as the front wheels is what stops you moving forward.

. If you still have a lot of pedal play adjust the screw or nut under the parking brake handle (righty tighty, lefty loosy) After that you should be good.  If it is STILL A LOT, THEN TAKE TO A MECHANIC cuz you flucked something up during reassembly, doode.  The parking brake is a wire cable.

I would have started to suggesting there's air in the lines, but if you're sure you bled the brakes completely then I'll take your word for it.  Beyond that, it's probably just a matter of the self-adjusting brake drums not being fully adjusted yet.  Try backing up and then pressing pretty firmly on the brake.  That should help the ratchet adjustment in the brake drums move towards the correct adjustment.  Alternatively you can remove the wheel and adjust the brakes yourself with a flathead screwdriver.