DA and MDA are different things and demand different techniques.
ILS Cat 1 is a Precision Approach, and so has a DA. You can begin the go-around at the DA (at the latest) so you will probably descend below the DA during the manoeuvre (I think by a max of 50'). That is OK.
Non-precision approaches (VOR, NDB, RNAV, SRA etc) have an MDA. You must not go below this altitude unless visual, so if you do not plan to fly a level segment at MDA then you need to begin the go-around a bit earlier, as you suggest.