If you are not confident in disassembling and you can't visually verify the back of your mobo, what you can do (for now) is this:
Lightly/gently get a feel of the stock cooler by "pulling" its frame (not excessive) just to get a feel if the cooler is indeed "clamped" in the mobo. You will easily know if it is not clamped correctly if one of the corners of the cooler can be "pulled" (has a certain give) than the other corners. Observe also the CPU temperature day-to-day if it exceeds the normal temps when idle.
If all goes well, then, you don't have to remove your motherboard in its mounting position. If you still have doubts, then call an expert to unmount some screws on the motherboard just to visually check the cooler pins behind it.