In our facility eschar is left dry and OTA, with offloading, LAL, etc. The eschar provides protection. Unless there is an underlying infection, which it sounds like may be the case with yours, with erythma, tenderness and warmth, whether there is drainage or not.
You might want to refer to a wound care center , get order for Doppler first to diagnose any circulatory compromise. Eschar will usually dry out and slough off, Iam currently treating one just like it. We first went to antibiotics with no success. No healing after 1 month. Doppler found a 70% blockage. No treatments you do will ever work unless there is oxygen to these tissues. In some cases surgery to create better blood flow is the only way. I hope this helps.