Would getting a new motherboard get rid of my ACPI problem?

I would sooner blame the PSU than the mobo. Personally I've had issues with PSUs not really sticking to standards (yes, it even said it was ATX12V v2.3 compatible) and causing startup issues. Luckily I've always used a backup PSU so I kinda managed, but you might want to consider that a possibility. Otherwise the mobo could def be a problem, but not necessarily.

Is your PSU a dual rail power supply? If it's single then it might not be pumping enough power to your PC. Having a dual rail 300w PSU is almost the same as a single rail 500w PSU.