Having read your post you appear to have 2 different problems with your bind off. The 1st is your bound off stitch count and the 2nd is the 'stair step' effect.
To solve the stitch count problem rember that every time you pass a stitch over that counts as a bound off stitch. This means K2 pass one over is one bound off stitch and doing another K1 pass one over means you have bound off 2 stitches, although you have knitted 3 stitches at this stage.
Because you know how many stitches you have when you begin a bind off row and how many you should have when you finish that row don't be afraid to count your stitches after doing your bind offs and before knitting the rest of the row. So, if you start with 20 stitches and need to bind off 3 stitches, bind off 3 stitches and count the remaining ones to make sure you have 1 stitch on your right needle and 16 stitches on your left needle. If you don't you can make your correction before finishing knitting that row. If you do you can finish knitting that row. Don't cut the yarn and pull through, when you have more knitting to do, you'll only have to rejoin it in and make unnecessary work for yourself. I also think this could be making the 'stair step' or jagged edge problem even worse.
When binding off over a few rows you can use different strategies for different situations.
If you are going to 'pick up and knit' that part of the garment later (eg do ribbing for an arm band or neck band I would cast off and carry on knitting normally as the ribbing will smooth out any jagged edging.
If you are binding off a part of the garment that is to be seamed, eg shoulder seam, you can done one of the following:
1) Cast off and continue knitting normally. Then seam to the matching piece, when finished.
2) Do the Sloping Bind Off as shown by Amy in the Binding Off section of the Free Videos. The instructions do seem a little confusing, but if you practise with a swatch 1st it will make more sense.
3) Check the instructions to make sure you are binding off the same number of stitches at the same point for both pieces and do a 3 needle Bind Off.
I don't think slipping the 1st stitch will help and, as you found, may make things harder for you.
Finally, (sorry to depress you) in the world of knitting patterns what you described is a normal bind off for most tops, which explains your problem finding an alternative.