> However, one of the new cells would still contain the "original" DNA.

This is where your question is based on a wrong assumption.

When a cell divides in 2, it is not the case that 1 daughter cell has the original DNA, and the other has the copy. Instead, DNA is made up of 2 complimentary strands, woven together in a double helix. During replication of the DNA, these strands are separated and a new copy is made against each original. Each daughter cell then gets 1 of those new helixes. Thus, each daughter cell has 1 DNA double helix, that consists of 1 original strand, and 1 new strand that was made during replication of the original cell.

Note that "1 helix" should be multiplied for however many chromosomes a cell has.