OK, completely re-writing this after re-reading the OP.
Let's start with the cause of odor in the underarms:
You have bacteria all over your skin at all times. Most of this bacteria is completely harmless. The bacteria particularly love dark, moist areas. Like your armpits. So, lots of happy bacteria are reproducing there to make more happy bacteria.
Then you sweat. You sweat more in places where skin meets skin as you have 2 sets of sweat glands involved. Sweat alone does not have a bad smell. However, you mix that sweat with the bacteria, and it starts killing off that bacteria, and then you have smell. More bacteria + more sweat = more smell.
To stop having this problem, you have several things you can try before going to see a doctor. But, honestly, I would start by going to a doctor, so you can get all the potential underlying causes ruled out, as some of them can be quite serious and require medical treatment.
First, you can try reducing the bacterial load of the skin. This means using an antibacterial soap/cleanser of some sort. There are a couple of excellent ones your doctor can prescribe. If you must try something at home before going to the doctor, then you can look up over the counter options on the internet. I would NOT try to use these on all of your body, nor is it an "every day, forever" type solution. This is a short term solution at best. If you have an objection to using antibacterial soaps/cleansers, then you are going to be washing several times a day.(with less than stellar results)
Second, you can try sweating less. Using an antiperspirant, not just a deodorant. And the formerly prescription-only strengths are now available over the counter. There are also other products an MD can prescribe or recommend that I just do not feel comfortable discussing in a forum. That would be stepping over the "practicing medicine without a license" line.
As to your original question. It is not the zinc oxide going rancid. It is most likely the lemon juice spoiling. If you keep this for more than 3 days at room temperature, or 7 days in the refrigerator, then that is exactly the problem. I would never use lemon juice in anything I needed to store. You might want to think about adding the other ingredients to a lotion multibase that already has preservative, and perhaps using citric acid to drop the pH.
