How to overide Admin password please?

Hopefully the Mac came with its original disks. If so, boot up with the restore DVD and erase and install from there. Shouldn't need the password.

Up to and including OS X 10.6 Snow Leopard Macs were supplied with (normally) 2 grey disks. One should say OSX Restore (or similar) and it should be indicated on the disk the version of the OS. These disks are machine specific - the grey disk from one model of iMac won't work with anothr model. The second disk contains the extra iLife apps.

If it is the correct OS X restore disk it should boot the machine and enable you to reset the admin password. I'm guessing the disk is either Leopard or Snow Leopard.

This link has instructions if you don't have a disk but towards the bottom the instructions if you do have a working disk.;
http://osxdaily.com/2010/08/10/forgot-mac-password-how-to-reset-mac-password/

To see whether the grey disk works try putting it in the optical drive and restarting holding down the C key and hopefully - if it is the correct disk - it'll boot. If so follow the instructions at the bottom of the link. If not, use the Single User Mode instructions.

One should say OSX Restore (or similar) and it should be indicated on the disk the version of the OS.

One's labeled Mac OS X Install DVD (only installs the OS and basic Apple apps, such as Safari and TextEdit) and the other is labeled Application Install DVD (installs other apps, such as the iLife ones, and has the Apple Hardware Test program). If the seller failed to provide them, get replacement ones from AppleCare.

Have you tried creating a new user? Does that exhibit the same problem?

I guess you can, but there doesn't seem to be much point as you said it works in Safe Mode. You need to create a new user in 'normal' boot to see that works.

Can't explain why it seems fine in Safe Boot and not in 'normal' mode, other than Safe Boot doesn't load everything (but I've no idea what it leaves out).

Did you actually do an erase and install of the OS, or just re-install over the top of what was there?

I'm afraid I'm stuck for ideas. More info about Safe Boot here:

http://support.apple.com/kb/HT1564

I'm wondering if safe boot doesn't load certain graphics drivers whereas a normal boot does. I'm just guessing here and may be talking nonsense, but it might explain the different behaviour.

Going right back to your first post, I wonder whether the seller booted into Safe Boot when he demonstrated it to avoid showing a likely hardware fault.

It's beginning to look like a trip to the workshop 😟

Just seen your edited post above my last one.

Did you do this to erase and install? Rrestart with the install disk in the optical drive holding down the C key. The Mac should boot slowly from the DVD. At the first screen chose your language and from the next screen from the menu bar select Utilities>Disk Utility......

For detailed instructions have a look at Pondini's excellent guide:

http://pondini.org/OSX/DU1.html

When you boot up it'll be as if the Mac was launching for the first time and you'll need to set it up as a new machine.