dude this is not a urgent situation

but ill give u some info on this...u are right that H1b for higher education is cap exempt so there is no time limit to file it and the processing time is quick...but for GC it truely depends upon the position you hold. Is is a EB2 or EB3 category? if it is EB2 it will take about 2-3 years...if it is EB3 about 5-6 years....and most importantly charify with your HR at your university if they have any intention on sponsering your GC. If it is a state university then they might have to justify paying taxpayers $$ to sponser a foreign national.

word of advice is that if they are reluctant to sponser your GC although you have a job for now(assuming you are in a state Uni) try to move to a private uni which are more progressive on these issues.....

good luck