If you want to go used, then its pretty pointless for others to suggest specific bikes since who knows if one of them is available in your area in your size. The odds are small at any given time.

So if you want drop bars, then yes a cx or gravel bike may be the way to go. I assume you carry stuff to/from work, so brazeons for a rack will be good. Something that can fit wider tires will be more comfortable on the pavement conditions you mention. More comfortable and possibly faster, actually.
High quality tires will make a huge difference with regards to comfort and speed. Riding thick heavy tires meas a lot of energy is being lost compared to low resistance tires, even if the low resistance tires are relatively wide.

10 miles in 45min means 13.3mph. cutting 5min off your commute will require you to average 15mph which is a significant jump in average speed. Cutting a few pounds off the bike could help, and being in a more aero position on drop bars could help, but combined that may not get you there and fitness will then need to play a part.