https://www.dol.gov/sites/dolgov/fil...nder-cobra.pdf
You have 60 days from the later of the date you're given the official notice or the date your coverage ends to elect to enroll in COBRA coverage
So in theory, if your new coverage starts May 1, you can wait to see what happens and if you don't need to use health insurance between the end of your old job and the start of your new one, you can save yourself $900. If you get injured April 30, you can still enroll in COBRA thereafter since you're still in your 60-day election period and be covered.
I don't like the fact that your old company hasn't done anything to start that process, though. Means if you need to elect there will likely be a ton of hassle to do so. But even if the time started March 28th, you're only 33 or so days out by the time your new insurance kicks in either way.