Brief Answer:
A muscle tear/strain more likely.

Detailed Answer:
I read your question carefully and I understand your concern.

Judging from your description the symptoms seem to be limited to the upper chest and do not spread more distally to the hand. That makes a nerve compression from the herniated discs you mention unlikely.
What seems more probable would be a muscular issue such as a muscular tear, with exacerbation from movements which require stretching and contracting the involved muscles. A couple of weeks therapy with a muscle relaxant like Flexeril and an anti-inflammatory painkiller such as Ibuprofen should help, although it should improve spontaneously.
Another possibility is simply pain radiation from the degenerative changes of your spine, which is addressed through physical therapy and Ibuprofen if needed.
So basically therapy doesn't change much from the one perhaps already recommended to you for your herniated discs, with the only additional recommendation of a muscle relaxant for a couple of weeks.

I remain at your disposal for other questions

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