Do I need to caulk the walls and around tub of my shower all at the same time?
In a perfect world you would want to but you don't have to.  The silicone is mainly bonding to the materials around the gap, which is generally your tub and walls.  It doesn't matter in that case at all.

The only point that matters is the silicone connectivity between your silicone and old silicone.  This is a very small connection.  On GE's website you can read that silicone does indeed bond with dried silicone.  It however says that bond is not as good as what you would get with two wet bonds - obviously.  But still works.  If your silicone was not all the way dry probably even better for you.

I wouldn't worry about it.  Silicone bonds well and lasts a while.  Maybe just keep an eye out for the area that you connected on.  But really no big deal at all.