Simplistically speaking an allergy will be a severe and immediate response. You should think of a person's reaction to a bee sting. A true milk allergy is very rare too. You would be looking for more severe symptoms such as rashes, facial swelling, and hives.
On the other hand lactose intolerance is an inability to process a particular sugar (lactose) so the symptoms tend to be milder. You will notice continual fussiness, abdominal cramping, bloating, diarrhea and/or vomiting. Also most babies tend to be less lactose intolerant as they age. Typically a young infant (3 months or less) may have lactose intolerance but by age one the baby will have resolved the intolerance.
At 10 months I would be surprised if your baby suddenly developed lactose intolerance. However, the symptoms you describe are generally consistent with what we see in our lactose intolerant baby. We notice if our baby receives very much dairy continual day after day, he has far more problems. Thus, we limit all dairy to minimal amounts. While he ate formula, we used a soy based one. He recently turned one so we switched to milk. After a brief trial run of two days on whole milk, he evidently displayed signs of lactose intolerance. We had to switch to soy milk. The results were almost immediate within one feeding. He lost his fussiness and stopped vomiting. The diarrhea and constipation took about three more days to resolve themselves.
I would consult your doctor again since a parent’s gut is a valuable tool. Good luck.