They do love to be held! A couple of things that might help:
I LOVE that you're using slings for your baby! It's so good for them and (especially when you have more than one child, lol.) so much easier!Your back shouldn't really hurt after carrying the baby in a sling. I don't know what kind you're using, but the main reason it hurts is because you don't have the weight distributed enough. Spread out the fabric across your shoulders/back--get someone else (hubby) to do it if you can. If you know an experienced babywearing mom, or can find a group, that can be a huge help. If you'd like to message me about the type, I'll try to help online!
Also, you might want to look into an exercise class for new moms, if you can. You have to be careful (get an instructor/class for new moms), but a little toning might help you feel better. Even gentle walk outside helps wonderfully!
Think about what's different when dad's on the computer and your little guy is happy in the bouncy. I found my two liked to be close enough to see me (within about four feet) and they needed to be good and awake--it's hard work to look around when you're little! They would last about twenty minutes (time enough for a shower!) When they were sleepy, my daughter hated the vibrate, etc.--it overstimulated her, but my son loved it and would nod right off.
Also, learn to talk a LOT :-) Your voice can be very reassuring and, believe me, you want to be able to comfort them over little stuff without running to them! I used to babble to my kids, or sing, or just talk about where we were, what I was doing, etc.
I know it seems like you'll never use the stroller, but every phase is a very short time in your life and they will soon sit up and want look around, and you'll be able to choose between slinging and stroller. (BTW, can he see you when he's in the stroller? And talk to him, and keep it moving!) I have to say, though, that I used a sling with my daughter until she was three. It's great at Disney--no stroller to haul--and on those walks where they WANT to walk, but get tired. My son, who is two, still gets sling rides for fairs, long walks, multiple errands, etc. I kind of miss it now that he wants to walk like a big boy!
Congratulations! Remember that nothing lasts forever and let me know if you'd like sling help!