My 5 year old daughter (who was my second child) had the same problem for the first few months of her life. I started off nursing her but couldn't keep up with her as she'd nurse, spit up, cry and then want to be nursed again to ease the pain of spitting up. It was a never ending cycle. There were no such things as burp cloths. I used bath towels instead. She also had explosive diapers so I knew there was a lot of gas in her stomach. She was a really unhappy baby and I was running myself ragged trying to make her comfortable. Her saving grace was that she was a good sleeper.
I made a trip to the pediatrician and saw one that wasn't our regular pediatrician. He suggested Reglan. (They told me if I did use Reglan to keep Benedryl close by because sometimes Reglan causes seizures and Benedryl would reverse it. Can you imagine using such a medication on a baby?) I'm so happy I never went that route. I did allow my regular pediatrician to put her on baby Zantac though. My husband has bad reflux and I felt safe using Zantac.
I talked with some mothers about it and they gave me the best advice. I switched her to formula and went with the soy formula. I think she had some sensitivity or allery to lactose. I also added some rice cereal to it to give it some bulk. It was a little thicker going down, but it also meant I fed her a little less often which meant less spitting up. That also might help with your constipation problem.
The other thing I did was kept her sitting upright most of the day. I didn't let her lay down much. I had her in either her bouncy seat or an extra car seat most of the day. I already was sleeping with her in the guest bedroom to let my husband get sleep. I put her in the bed next to me and allowed her to sleep on her stomach (I spread out a towel underneath her just in case.) I know that stomach sleeping is frowned upon due to SIDS, but I was sleeping right next to her and I'm a light sleeper and don't move during the night.
All of that got us through that difficult time and she just outgrew most of it by the time we got to solid food. She's now 5 and we still talk about that time. I don't have as many baby pictures of her as I do my son because she was so unhappy, I was so worn out, and she always had spit-up on her. But she's happy and healthy and that's what counts.
Good Luck and you will get through this!