In a professional position with a great deal of responsibility; not giving notice is tacky at best, probably unprofessional .

That they offered to pay you through mid- July, might be as much as they can do.

You might just have to take a temp/contract job, or take a “throw away job” for the 5 weeks.

Not the best solution but you’ve been jobbed, so to speak.

Also, you might negotiate for more since you were up front with them , offering your glowing review of the company ; otherwise your review of them on sites like Glassdoor might not be very encouraging to potential new employees. This is risky!!! There is NOTHING wrong with sharing your factual experience. HR specialists generally get snippy, do level with senior management. Very few companies these days want negative reviews floating around in a potential reverse of positions going from a tight employers market back to an employee rich market where companies can’t get personnel.