Just to check - does it mean something like "I'm so touched, I'm about to start crying?" thanks.
To me, it's when you can feel the tears and are pretty sure you're going to start crying.
I'm pretty sure this is a fairly recent development. As long as I've lived, you could 'feel tears welling up', but it's only in this century (in Br Eng) that you could leave the tears out and just say "I'm welling up". The rather rare word 'to well' originally meant '[of a liquid] to rise above the usual level' if you overfill a cup with water, the water wells up over the rim.
But tears are now the default 'weller'!