I like the advice on cutting down a little on the formula. It could be that he is simply not very hungry, and therefore not willing to eat the solids.
If you take the time to figure out WHICH solids he prefers right now, you can serve them on a consistent basis until they become part of his eating habits. You should avoid fried foods, they aren't appropriate for a one-year old. Offer him "rainbows" of foods, start off simple: canned peas, canned carrots, avocado, mashed or steamed sweet potatoes or yams, mashed or small-cut bananas. BUT, don't try too much at once, just one or two foods a week consistently.
For the constipation, offer him some applesauce or prune juice.
If you want the best for him, offer him non-processed vegetables and fruits that are soft enough for him to eat. At one year, he should be able to feed himself small pieces of foods, well-cooked or soft enough to mash with his gums. Avoid foods that are "prepared" or "out of a box" his body needs to learn healthy foods that aren't processed.
Have you talked to his pediatrician about nutrition?? Our pediatrician had some great advice, even some great handouts when we transitioned our daughter to solids.
Offer him some creamy oatmeal too, good fiber source for his system. Start off with the Earth's Best (baby food) oats, mixed with formula (or whole milk). You can add something he likes to it for flavor, like mashed bananas or something.
Our pediatrician encouraged us to be off formula by one year and on to whole milk and solid foods. Our daughter stopped taking a bottle at 10 months, got formula in a sippy cup from 10-12 months, then at 1 year we transitioned to just whole milk. That was with a lot of preparing from 6 months on to make sure she ate lots of solids. Talk to your son's pediatrican, it's totally worth making the appointment, and see what he/she recommends for him since he is past a year already.
Best wishes!