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If I clip off these flowers when they are dying off and bury the dying
flower clippings in the planter will they grow a new? I thought I had
read something about this on here recently, but wasn't sure if these
two types of flowers fall into that category.

Removing the spent flowers is call "dead heading." The spent flowers will
not grow new plants. If you look at zonal geraniums (the kind sold as
annuals that are actually pelargoniums) you will see that they have nodes
along the stems where new leaves emerge. If you bury stems (not spent
flowers) so the node is below the surface, you should get root formation.
To improve your luck, you can use some rooting hormone. I haven't heard of
anyone rooting marigold cutting, but I guess you could try. You will can
usually find marigold seeds at garden centers. I think you would have
better luck trying to grow marigolds from seed rather than cuttings.