To find out how much ram you have, go to the apple icon at top left of menu bar & choose 'About this mac'. Also specifics about your mac can be found if you click oon the 'More info ...' button.

Your ISP is Cox who are known to block individual users from using p2p or specific p2p programs. Check Cox listing here ISP's known to BLOCK or Filter p2p or specifically Limewire

After so many years, did you try deleting your LW prefs file/folder? Before you try it, make a note of all your settings in LW because once you delete your prefs you'll need to reset all your options next time you open LW. LW MUST be closed when you delete the prefs. You'll find them here: /Users/your_username/Library/Preferences/LimeWire
And send them to the trash. If this doesn't help with your connection then see below.

Try this ... change LW's listening port. Go to LW's preferences, Advanced>Firewalls & change the listening port to one between 49152 - 65535. A good one to try is 64049. But also try others if that doesn't work. After you make the change, close LW & re-open it. I hope that has some positive effect.

BTW did you get LW 4.8.1? If not, send me a personal message (PM) if you want it. It will be a better program than version 3.x. I believe 4.8.1 was better than any earlier 4.x version & better than 4.9.x
I never did try version 3 but did use version 2 for over 2 years in OS 8.5 before upgrading to 4.0.4 in OSX.