Hi Jason,
The CMOS battery on the motherboard would have nothing to do with the
physical starting of the HD's, floppy,and CD-RW drive.
As you certainly know, all of the aforementioned drives contain motors,
which have bearings.  After time, these bearings can become dry.  When this
happens, these motors require more and more current to get them going,
taxing the power supply.  It's *possible* a bearing in one of the drives is
getting dry, but unlikely.
Have you recently added another drive?  That would certainly explain the
trouble, especially if it coincides with the installation of a new drive.
I think your guess at the power supply is right on the mark.  I would
replace the supply with a new unit with at least a 300 watt rating, but get
even larger if you can afford it.  That will allow for even more expansion
in the future, should you desire it.
Even though computers *do not* draw that much compared to many appliances,
that initial surge  when you first switch  them on can be several times what
they draw after they are up and running.
Try not to skimp on the power supply.
HTH,
Curt