I'm new here and also a new owner of a couple of old lt80's.....so don't listen to me too much!
Sounds like the started is working, but not the solenoid. Not sure why that would happen so quickly to a new start. I'm sure someone will chime in with ideas for you. Just curious....how bad of a job is it to change the starter on these little quads? I just tore into the clutch on one of mine over the weekend (actually got to take it apart like 3 times!). That first time was horrible. I don't think anyone has ever done anything to this quad so it was really hard getting that clutch cover off. I now need a couple of new allen head bolts to replace the ones that were abused trying to get them loosened.