I have a Sony VCR. Model SLV-N51. I record a lot of television shows on new Sony VHS tapes and when I play back the tape the picture scrambles every so often. When I rewind it back to where the scrambling begins it will sometimes play back clearly. Also the sound either gets really low and then back to normal. I used a Vcr Head Cleaner-dry type and it did not help. Has my VCR gone bad?

First of all I want you to check the tape inside it's case as soon as the problem occurs. Stop it and eject. You'll find a button on the side of the VHS tape, press it and you should be able to lift the flap. Look at the tape it should be smooth and shiny. If it is there's nothing causing damage. If you see any damage then the mechnism is causing it.
If that is good (which I think it will be) I want you to take off the top cover of the VCR and with some Servisol Video 40 spray, blast all the surfaces that the tape comes into contact with. You can wipe them too with cotton buds. Give the heads special attention. Once done leave for 30 mins and try it out.
I think the problem might be dirt that the head cleaner tape couldn't shift. The sound dipping being the hi-fi sound dropping out reverting back to the mono sound head. When you rewind the video tape acts like a cloth for an instance, hence the better performance.