I can confirm that there *are* a few routers that will simply become botched after trying to handle too many connections / too many *weird* connections and the symptoms would just NOT go away after closing the "offending" softwares (even if they're not doing anything intrinsically wrong). This is because:
1) Your IP address + port have been broadcast to many sources who still keep trying to connect to you even if you close the software (this is half of the reason why it is "solved" when you shut it down - you get a new IP address from your provider when it's back up);
2) The traffic "deluge" (as your router sees it) triggers a bug that botches it. I've seen quite a few wireless routers drop dead that way, especially their wireless adapters. One funny case had its Ethernet ports working while its wireless function needed a reboot to work again.
So get ready to see your router as the culprit here. I recently got a new cheap router that seems to have a bit of trouble handling my home setup - it's a shared student house with 12 devices connected at the same time (at top usage hours), half of them being heavy users. We were consistently getting hangs with another router, so we're trying another. This one didn't hang (yet), but its web interface gets really slow when too many people are connected, a few status screens simply do not show up and I've seen it reboot itself (probably through some sort of internal watchdog software) at least once. It's better than a full-fledged hang, but you see what I mean.
Steps to either confirm it or dismiss it:
1) Install other P2P software and test it (but hey, don't ditch Shareaza, we're cool...
). See if you get the same results after a while;
2) Try going to Settings - Networks - Gnutella and choose Mode
Leaf instead of Optimal or Hub. You should have fewer connections that way and it's a good way to test it (the best option overall is Optimal, because you're helping the Networks themselves, but it's a good way to test it).