Running shoes are rotten cycling shoes. I can say this with authority because that's what I've been using all week, although I haven't gone further than 20 miles and my feet have mostly been OK in them.

Check out the bike I've been riding too (this is a crappy beach rental bike!) I did manage to average over 17 mph on the local bike path yesterday, so I'm thinking I may need to trade in my roubaix. It has funky oblong chainrings, the crank arms are too short and the frame is way too small - I don't feel stable standing up on this thing but on a perfectly flat road I can get it going.

But yes, stiffer shoes for you. My wife used to use hiking shoes for cycling, that worked out OK.