Just to be clear, I can leave Michigan City in the morning and catch my limit of Coho along the beach in Indiana waters, return to shore, swap out coolers and then take another limit across the state line in Michigan waters with my Michigan license and then return to Michigan City?
How many fish would I be allowed total in possession if I did this two days in a row?
Yes, correct you can catch an Indiana limit, return and swap coolers (if you want to be really solid, take a pic/write a description of where caught), and then you can launch from Indiana waters, travel to Michigan waters, catch a Michigan limit on your Michigan license and return with them.
Regarding possession limits, this is what the Indiana regs say -
The possession limit is two times the daily bag limit and does not apply to a wild fish that is processed and stored at an individual’s primary residence.
Assuming you cleaned and stored those fish at your Indiana primary residence, there would be no possession limit issues.
Here's Michigan's wording on possession limits - last two sentences are particularly relevant, and why people are always advised to not have their fish from another state with them on the boat. Otherwise you're gonna get a ticket for overbagging
Possession Limit: In addition to 1 day’s daily possession limit of fish, a person may possess an additional 2 daily possession limits of fish taken during previous fishing days provided the additional limits of fish are processed (canned, cured by smoking or drying, or frozen). This provision does not apply to lake sturgeon or muskellunge. Anglers who have not attained the age of 17 are entitled to the possession limit even though they do not have a fishing license. A person fishing waters bordered by other states or provinces AND possessing multiple fishing licenses may possess the limit allowed for ONLY ONE license while in transit. When fishing in Michigan waters, Michigan possession and size limits must be followed.