2005 Dodge neon sxt 2.0L positive battery cable burned.?

I recently noticed that my positive battery cable was burning. I also noticed that the negative terminal was completely corroded through, So i have replaced it. Since i haven't noticed any more burning smells. Should i be concerned. Can this cause the cable to burn up? Also today i took car to get oil changed and they said the battery was leaking? Is this a problem? If so what caused it? If anyone can help me i would greatly appreciate it. As i am driving to Utah today. Will it be safe to drive 600 plus miles?