Getting bigger could be due to milk being drawn into it and unable to escape. It may go down a little again after the feed. You really do need to shift it, otherwise your duct will start to get inflamed and you'll get this feling of tenderness tracking back in a wiggly line through your boob 
When the stuff does come out it might be gritty, toothpasty, yellowish (like a squeezed spot) etc. There may also be more than you expect from further back in the duct, depending on how long it's been blocked.
Feeding leaning forward, or dangling boob in bowl of warm (not too hot!) water might help -i.e. gravity.
Some people wander round with a ball of cotton woll soaked in olive oin in their bra to try to soften the skin before the next feed.
With the needle thing... it's not strictly advised to do it at home because of the infection risk (though arguably a boiled needle is less of a risk than a fingernail). If you do read up and decide to give it a whirl, then you only need to make a tiny prick in the skin in the middle, not jab it in deep or anything . People I know who tried it say they barely felt it, and afterwards were able to express the gunk.
Sometimes the blasted things seal over again, so they can go up and down a bit.
Also, if there's a cause of this (and maybe your mastitis too?) it would be good to try to work that out. Both result from less than ideal drainage of the breast and often suggest there's something that can be tweaked about the way the baby feeds.
Good luck!