Somebody else is having the same problem with the latest Beta version & has posted in the developers' forum. Check out the thread here (click on link): LimeWire Forums

Try disabling TLS as suggested in that thread & see whether that helps at all.
You could also reply to that thread if you like, so that the developers know that the issue exists in 5.3.6 & for Windows users as well. You'll need to register at the other forum in order to post