If you are gaining one stitch each row, my guess is that it might be happening at the beginning of each row. When I first started knitting, working only in the knit stitch, as I got to the end of the row and turned my work, my yarn would inadvertently end up going over the needle, and the last stitch on the previous row would look like two stitches when the yarn was going back over the needle instead of staying in front of the needle. This made me 'see' two stitches, and then I knit both of them... so annoying... I don't know that this is what is causing your scarf increase, but I can admit that I did that for awhile before I figured out what I was doing with my yarn...
Since you are making a scarf, you probably don't have too many stitches to start with. One approach would be to place a marker every 3, 4 or 5 stitches, across the row... just so that you know exactly how many stitches are supposed to between each marker. Then when you do the next row, you will be able to identify exactly
where you are picking up the extra stitch, because you will have one extra stitch between the two markers where you are getting the extra stitch. If you can isolate where it is happening, that could help identify where; and a picture would help too.
Don't give up!
We'll get to the bottom of it!