Try to think like adcoms at the Ivy league universities where you shotgunned plus G’Town and UChicago – keep in mind you applied once and they already chose to deny you and keep in mind they have three times the number of highly qualified applicants/places:
Highly selective schools don’t go “Oops, we really missed a diamond in the rough”. Admitting you a year later would be like admitting they made a mistake, right? They’re sure they did things right the first time, so if they didn’t admit you the first time around, well, when you reapply, are you trying to say you know better and they should have ? 
They admit perhaps 5 or 10 or 15% of their applicants, of whom they got a fresh batch admitted. Transfers will replace the places left by the very few freshmen who decided to leave. They’ll want new applicants rather than someone they already decided didn’t belong. They will give you another look but trying to see anything that will disprove their earlier diagnostic, so it’s not easy.
So, legally, yes, you can send them an app and the app fee, they’ll take it, they will pull up your previous application too along with the comments the adcom made, perhaps the same regional rep will read it and remember you, wonder if you did anything extraordinary that’d prove them wrong in denying you the first time, and unless you cured cancer in a year, yeah, they’ll validate their first decision, ie., that you shouldn’t be there and there are better qualified candidates.
That being said, it means that there are lots of good universities where you haven’t applied 
where you may have a shot in January 2022. Just remember that if you apply then, you won’t be able to re apply to those if that attempt fails.
After 3 semesters you’re further away from high school so you’ve had more chances to distinguish yourself and your odds are a bit better, but for the Ivies you applied to, the odds are absolutely minuscule and, as I mentioned twice already, they primarily either look for CC or lateral transfers (to summarize, go upthread for details). However, in January 2023, if you have excellent grades and activities, Northeastern and NYU are open again as well as BC since you were WL not denied. (UCLA will prioritize California CCs before someone from OOS but you can try, too).
You can also try for Vanderbilt, Emory, UVA, Wake, Tulane, Villanova, UMD College Park. I think it might be easier to transfer to these from GWU (it’d be more of a lateral transfer since everyone knows how good GWU is for poli sci) although probably only if you chance majors or something (because everyone knows how good GWU is for poli sci so they’d wonder why in the world you want to leave.) Rutgers-> UMD College Park for Poli Sci would make sense, too, but GWU->Villanova for poli sci would be strange.
Finally, among your 3, don’t discount Penn State without visiting: it’s less good than GWU for poli sci, but it holds its own and has unique concentratons that could interest you (including SODA*) the campus is nicer than Rutgers, their career fair is powerful, and you could be in Paterno Fellows (Honors) which would lead to a qualitatively different experience.
*https://soda.la.psu.edu/undergraduate/prospective-students
To be clear, I still think GWU is your best choice but you should consider your choices carefully and review them before you dismiss them.
Rutgers:
At GWU, were you admitted to Columbian (Arts&Sciences)?
