My son had very similar sleeping habits at that age and didn't settle into longer naps until he was 7 or 8 months, maybe, I can't really remember when things changed (that should give you hope ;-).
He did sleep very well in his car seat (I would take a long drive across town to the grocery store and then shop, and he would sleep both directions) and he really loved sleeping in the swing -- just make sure you have a swing that plugs into the wall because batteries die. He loved being held (would sleep in my arms while I watched TV or read a book) but hated being worn so that never worked for us.
We also had a bassinet topper for our stroller and he loved riding in that, so I would literally walk him around the house in it, delivering laundry to bedrooms, going to the bathroom (ahem), pushing him back and forth while I ate. And of course we took long walks around the neighborhood with either the car seat or the bassinet attached to the stroller. Overall he really liked motion. Might want to give it a try.
In the end, I just had to be consistent in laying him down before he was exhausted so he could learn to sleep on his own. And we did a modified cry it out plan, because I couldn't handle letting him cry endlessly. Basically, I would let him cry for five minutes at a time and then check on him, hold him until he calmed down, if needed (sometimes he just needed my hand on his belly or to hear me singing, while he was laying down) and then go for another five minutes. It took almost half an hour the first night, less than 15 minutes the next, and then 3-5 minutes after that for a few weeks, and then he just stopped crying altogether. I think he needed to learn that my husband or I would come when he really needed us, and now he only cries when he's actually teething or sick, which is great. But we didn't do this method until he was about 8 months.
I hope this helps and that you can get some relief. And like some other Mama's have said, don't worry about the housework, unless it's something that bothers YOU. I'm a neat freak so I understand your pain and simply have to keep a neat home, but it's not always so clean...