mpops, it sounds like you've got a really good appreciation of your LO's sleep cues - well done you! It's really hard when you can see how tired they are and yet they just won't drop off. Is there any activity that does normally make your DC drop off? DS's used to be feeding, going out in the pushchair and going in his car seat, oh and he also used to nap if I rocked him in his rocker/bouncy chair thing. I figured that it didn't really matter where he was napping, just that he was getting some sleep.
DS never fitted into the whole 3 hr feed, awake sleep cycle thing, so I just went completely by him - fed him when he was hungry, let him sleep when he looked tired etc. This saved me a whole lot of stress in the long run and DS seemed much happier. This was after a night when DH quite correctly told me that DS was crying because he was hungry and I was adamant that he couldn't POSSIBLY be hungry because it hadn't been 3 hrs like all my friends babys' were doing. 
Um, yes, he was hungry, very hungry. Ahem. Moving swiftly onwards....... Oh and the first time that he ever slept in his Moses basket at night for 45 mins was at the age of 11 weeks, he co-slept with us for the vast majority of the time.
At 13 weeks, you are your DCs world and the only 'activity' that they need to be doing is being near by you and watching what you're doing. Have you tried just sitting/lying and looking at each other? No words/movement necessary but you are the most fascinating thing that your DC has ever seen 
All the baby classes etc at this age I'm sure are just about getting Mum out and about, all the baby wants is to be close to mum. Remember that the little sponge brains that they are, they're absorbing info like no-ones business, even stuff like, how does mum's face work, how do these funny bendy things attached to my body work and WOW look, FINGERS and, and TOES!!!
As regards fiddling about in the kitchen whilst not getting food, have you tried writing yourself a list of things to do? I did this and it consisted of things like: going to the loo (tick!), put socks on (tick), get bowl of cereal (tick), get glass of squash (tick). All very little things normally but huge tasks when you've got a LO and I used to feel so much better at the end of the day when I could tick some things off. Note that there is no mention of housework or anything else. When you are on your own, your only task is to look after your DC and the way that you do that is by looking after yourself as well.
Can you fill your fridge with easily grabbable food? I used to have a bowl of cereal for lunch sometimes, poss not the most nutritious but it was filling and did give me calories, calcium, fibre and vitamins and minerals. Also having things like quiche around, so basically open packet, tip on plate, eat.
Am thinking of you lots and hoping that you're having a better day today x