I suspect part of m'board is dead.
CMOS battery supplies power to CMOS memory and RTC (real-time clock, which is usually made my Dallas Semiconductor). BIOS chip is permanenetly or semi-permanently written (semi-perm is flash BIOS). Default-BIOS loads to CMOS memory, then you make changes in BIOS (which is really changes to BIOS loaded into CMOS memory) and is saved and held in CMOS memory by battery.
CMOS checksum error means CMOS memory is not operating properly. Same results with new battery means nothing is being saved to CMOS memory.
Does BIOS retain any changes made if you don't turn the computer off? Make some changes, then do a warm boot, go back to BIOS, if changes are still there, and if you only lose changes when computer is turned off, battery hold-up circuit is dead.