First off, order a soldering iron with a smaller tip, preferably with a beveled tip, rather than
a conical tip. It's worth the price to spend $50 and get a heat-controlled, adjustable unit.
Also, even with the fat tip, you should be able to reheat all the solder joints that overflowed,
and draw off the excess solder. Once the solder has re-melted, pull the tip off to one side
of the joint, and the excess solder will follow. Surface tension. May need to repeat a couple
of times. It's not too difficult to do.
Be sure to wipe excess solder off of the iron tip on a damp sponge in between every single pass
at the board - this is very important. You also need to do this during normal soldering, or solder
globs will build up on the tip.
Do not power up until every solder joint is fixed and inspected.