If you can see your stitches, they're too long and/or the thread is too thick.
Paper piecing requires 20 stitches to the inch, so the paper is well perforated and you don't distort the stitches when you pull the paper away. They should be so small that you can't see them in the well of the seam.
I don't match the thread to the fabric. I just use a neutral dark or light thread, usually grey or tan, depending on whether my fabircs are light or dark, and never have any problem with thread showing.
Fine thread, size 50, is ideal for paper piecing. I really like Aurifil, but there are lots of cotton threads out there in size 50.
The seam will also feel thicker to you because there are two threads in there, rather than one, as you have in hand piecing. That's another reason fine thread is important.
No, you do not need the adjust the tension. You want a nice, balanced stitch.
Does this help?
Anne in Vancouver, Canada