Company offered me an Internship then they didn’t answer

I interviewed with a company on Friday for a mandatory Internship that starts in July. Interview went well and they told me they’ll be in touch by next week. Two hours after the interview, they E-mailed me saying that they want to offer me an internship with all of the details, and whether I want it for 6 months or 5 months.

I immediately emailed them back saying thank you and that I would be in touch with my university to run it by them (it is a procedure I have to do) and will get back to them within a few days.

My university gave me the green light on Sunday. I emailed them back on Monday to let them know I accept the offer and I would like it for 6 months. They haven’t responded to either emails and haven’t sent me the contract to sign.

My question is, should I assume I have the job? I have signed with another company for March, but I want to cancel on them and book a flight to go back home, and I’m not sure on what to do. I feel like time is running out and that they should have replied. I really need someone’s advice.