I ended up calling back the day afterwards (last wednesday) and I just got a call back from the hospital's HR this morning. The residency program has different start dates and their last start date for the summer is in the second week of August (Just barely giving me enough time to retake due to NCLEX's 45 day period).
I still have a chance to make it, (The HR lady will be talking to the hiring manager of the floor I was going to work on) so fingers crossed. If it doesn't work out, HR said that I could join the fall cycle for the program and start in October. If interested I'll keep you posted with what they officially say. I hope things are going okay for you and hope you don't get dropped!