I would give PC mode on your TV a closer look. From what I’ve read, PC modes generally have comparable or lower input lag than game mode. I would think the washed-out colors are either due to a lack of calibration or, if it is calibrated, then those colors are more accurate.
Side note regarding using the Dreamcast with the OSSC, the typical recommendation is to use a sync combiner and connect the Dreamcast to AV1, as AV1 has low-pass video filtering that the Dreamcast can benefit from. (AV3 only has low-pass filtering applicable to 960p/1080p/1200p.) Personally, I use the SCART output from a BeharBros Toro, but the Dreamcast SCART cable from Retro-Access is a really good option if you have no use for the additional functionality (scanlines, brightness boost). If you already have Toro, you can get a DE-15/VGA to SCART cable (or adapter) and toggle the switch between the DE-15 and SCART outputs, which will get you RGBS from the DE-15 port instead of RGBHV.