Your testing something wrong. If you have 12.75 across the posts of the battery, unless the battery went bad, you will have at least that or more when it's running.
With it running WOT, check the voltage across the battery. It should be above 13V, but not over 14.5V. If it's below 12V, you likely have a charging issue. Check your battery connections. However, if there was an issue there, you should be having starting issues. Don't get excited about replacing parts yet. You don't even know where the issue it. Don't goof up testing the system by swapping parts all around. That just makes it hard to diagnose.
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