It seems that biology books refer to DNA as 'a' molecule (implying that it is a single continuous uninterrupted double strand) ... but in those same books chromosomes are depicted as 23 separate objects (chromosome pairs) containing DNA. Are these illustrations misleading ? Are chromosomes somehow connected end-to-end or are they disjoint ? Is it a single noodle or is it 46 pieces ?
Unbroken strands of DNA are single molecules. So Yes, DNA is a molecule. However, the sequence of nucleotides allow for a near infinite variety. So unlike other molecules, like collagen, or water, they cannot be characterized with a unique descriptive name, other than by their sequence. I think this is the cause for confusion, because in this sense, DNA is a class of molecules, and we don't generally give them names. Each chromosome has two unique DNA molecules in diploid organisms (except in rare cases of highly inbred individuals where the two molecules might be identical). So if you have 23 chromosomes, you have 46 DNA molecules in each cell nucleus.
When a book talks about "the DNA of a cell" it is talking about all the DNA molecules which are contained in the cell.
The DNA in a cell exists as separate objects called chromosomes, each of which is usually described as a single long molecule of DNA. (The long molecule has two strands, and coils up around many copies of special protein molecules.)
The chromosomes of a cell are separate, and do not have DNA running between them.