I like that the puppets looked at each other and your mouth lip sync and the amount you opened the mouths were appropriate too.  It is hard thinking about your other puppet while you are doing two at the same time.  You don't want the non singing puppet to distract from the solo but it helps to make small movements so they still look alive. When I have two on it helps me to make them move the same, like swaying or nodding together. Keep it up.

So here's video 2. Ive added a simple white sheet hung for a background, lowered the camera angle, added a front light(I think that I still need another light for the background just to get rid of the cast shadow, but I thought the shadow gave it a little character anyway.) and also upped the quality. The articulation of the dogs mouth(even when using my dominant hand) is much less than that of the lion. So now Im working on another puppet off the same base as the lions. Hope you enjoy!
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https://youtu.be/z_u-clNzu6U

Also, thanks for all the comments! They are appreciated!

Yep you are getting there!   Your sync is good but I do have a observations. First try to drop the top of the head a bit so the puppet is not looking up so much.  Part of what is causing this also is what is called "flip top". This is when you are moving the top of your hand more then your thumb when you talk with a puppet.  It takes practice but try moving the thumb only to open and close the mouth. This leaves the top of the head and they eyes in approx the same position so the eyes are pointing forward. Like I said it takes practice and a lot of it.   Don't be discourages though you have a really great start.

I agree that if you light your background also with a light between your puppet and the background it would help. I kind of like that the sheet is moving in this video, it gives some nice movement but it could get distracting for some performances.

I would suggest watching some Muppet or professional puppet shows and look at their camera positioning. Try to mimic what they do. If you are going to use a fabric background, consider using a curtain rod to get even pleats or make it totally flat.  A white sheet looks a bit like he's standing in front of a clothes line. Lighting is a challenge and shadows can be interesting so keep it up.  LOL, The animals "Rising Sun".  I bought that album when I was very young. You have nice puppets.