Doxorubicin is fluorescent in the orange/red channel. No way to double stain with PI even after extensive washing, since the drug is retained in the nucleus in a dose dependent way.
Not sure if you want to make cell cycle analysis or live/dead assay. What I am writing it would apply only to the second case.
The only solution is to use alternative dyes. The SYTOX green DNA dye should work since it is a green emitting dye with excitation in the blue, so with the same laser used for PI. You should not get spillover of the doxorubicin fluorescence in the green channel.