The high concentrations of proteins crowding cells greatly influence intracellular DNA dynamics. These crowders, ranging from small mobile proteins to large cytoskeletal filaments such as semiflexible actin and rigid microtubules, can hinder diffusion and induce conformational changes in DNA. While previous studies have mainly focused on the effect of small mobile crowders on DNA transport, we examine the impact of crowding by actin filaments and microtubules. Further, because actin filaments and microtubules are formed by polymerization of actin monomers and tubulin dimers, respectively, we also investigate the role that the polymerization state of each protein plays in DNA transport and in the time-varying conformational changes of single DNA molecules diffusing in in vitro networks of polymerized and monomeric actin and tubulin. We find that crowding by actin monomers slows DNA diffusion while tubulin crowding actually increases diffusion coefficients. Monomeric actin crowding DNA diffusion, more than when actin is polymerized, while crowding by tubulin dimers increases DNA diffusion more than when tubulin is polymerized (microtubules). Further, we find unexpected relationships between DNA coil size and diffusion when crowded. All crowding conditions lead to some degree of DNA compaction, but less compaction enables faster dynamics.