No. Since you can do both - 'sing and dance' - doesn't necessarily mandate you MUST be doing them both at the same time. What if, however, you could not dance without singing? In the case of bearing arms, the ability to keep arms is a prerequisite to bearing them.
Considering them separate Rights implies the ridiculous notion that you could carry something you didn't have the acknowledged right to have in the first place!
IMHO, that's also why they used the unquestionable joining conjunction 'and' instead of the refining/exclusive conjunction OR.
I don't agree that "keeping arms" is a prerequisite to bearing them.
Many anti-gunners adopt the view that all of our arms should be KEPT in central armories and only released when we can evidence a need to bear them.
I would also point out that the word "OR" is more ambiguous that you are stating. That is why logical arguments distinguish clearly between "inclusive or" and "exclusive or".
"Inclusive or" is logically equivalent to "A or B or BOTH A AND B". "Exclusive or" is logically equivalent to "A or B but not BOTH".
You are correct to consult the context to resolve ambiguities. For example, I doubt very much that our Founders wished to protect the keeping of arms and the bearing of arms, but not the manufacture, sale, purchase, transfer, modification, and storage of arms, as well as the training in use, inheritance, gift, and any other activities associated with being an armed people.
For proper interpretation, one need only look to the First Amendment and the very strict scrutiny afforded activities that are at all similar to exercising "freedom of the press".