What could cause the urge to have a bowel movement after eating, also constipated?

Normal. There is a gastro-colonic reflex which can cause this urge to have BM shortly after eating. Likely normal and good especially if constipated! Try to maintain a balanced diet with plenty of water and fiber.
Defecation. The urge to defecate after eating is due to the gastrocolic reflex. There is with stretch in the stomach and GI release of neuromodulators with motility change in the colon. Constipation likely lack of adequate hydration and low fiber. One should have about 2-3 liters of fluid a day and have foods high in fiber including oats, brown rice, avocados, kale, strawberries. Discuss with a doc.