The below site contains logs from the USS Constitution as well as journals from crew members that give you an individual perspective of engagements. As David M said these type of documents are where you might find the details of individual engagements.

https://thecaptainsclerk.com/

There are links to archival material from the museum collection as well.

https://archive.org/details/ussconst...t=-week&page=2

Lots of stuff if you really want to get into the weeds of the War of 1812!

I found a nice volume on privateers and privateering in the archives that does have some descriptions of engagements. (can't find the link atm).