What should you do in relation to this patent?
First, talk to a lawyer.
Then, you should obtain a copy of the patent, and determine whether (1) the patent is still in force; and (2) whether your product infringes the claims of the patents. If the answer to both is yes, then you would be liable for damages if you used, import, manufactured or sold your product in the country of that patent.
You mentioned US 5972360. This appears to still be in force, and will be until 3 September 2018. The independent claims are:
1. A self-tanning product in the form of a towelette comprising a towel impregnated with a self-tanning composition.
7. A self-tanning product in the form of a towel impregnated with a self-tanning composition containing an effective amount of dihydroxyacetone and ethoxydiglycol and enclosed in a liquid-proof packet which can readily be opened for one step application to the body and/or face.
If you make, sell, import, or use either of these in the US without a licence, you will be infringing the patent. Assuming the patent is valid, you might therefore be liable for damages for your acts.
You could therefore look to obtain a licence, or just wait until 3 September 2018 when the patent expires.
Should I add the same patent number to my packaging?
Absolutely not. The patent number there is added to put others on notice that the products sold are covered by the patent, so that the patent owner can maximise their damages from infringement. It would be very much an own goal to put the patent number of your own packaging.
Why do different manufacturers mark their products with the same patent?
They probably subcontract the manufacturing to the patent owner, or are licensees of the patent owner.