All the adapters I have tried and use have full handshake support. (Let me clarify.....support for the DB9 RS232 Control Lines). Do note that the support for the lines becomes a USB packet transfer issue and so there can be more delay from application to actual end of cable than when an actual resident COMM port in the PC is used.

I was using a USB to RS232 cable to in turn control an RS232 to RS485 converter. This RS485 converter used the RTS line to control the drivers on the converter out onto the half duplex RS485 bus during transmit time. I would have a VB6 program command the RTS on and off and measured a 6 to 15 millisecond delay from the time of the VB set RTS command until the driver out at the adapter became enabled (or disabled). For many applications this additional delay is practically transparent. On the other hand any application that was trying to use the handshake lines as a signalling mechanism the delay can be a huge problem.

Michael Karas