Replacing a broken sprinkler head with the same head should not cause your pressure to go down, unless the tee is also broken or there's another leak somewhere in the system.
However, none of this would explain why it works okay for 3 minutes and then stops working. Your zone valve may have to be cleaned out, or maybe your clock is malfunctioning, if you have one.
Another possibility is that you got dirt or rocks in the line when you changed out the broken head. Somehow, this debris keeps clogging and then uncloging the line. Take off the new head and open the valve to clean out the line. Put the head back on and see if the pressure still drops after 3 minutes. If it does, then the problem is probably in the valve, and you'll need to take it apart and clean it out.