It's so nice to know we're not alone, isn't it?! DD2 is 8 weeks old today, and has had exactly this same pattern - feeding constantly from late pm to mid evening, before conking out (normally on the boob, much to my dismay, since I worry about setting up the wrong association etc. etc.)
DD1 didn't do this at all - used to do really big feeds and then settle herself to sleep, so I was really concerned when DD2 started doing this.
I would really echo Mears's advice though - getting them to have a really good feed from one side first, is key. I've discovered, for example, that dd2 being so windy means that she often seems full, when in fact she's not, and she just needs a burp. I've been surprised often that when I put her back on the same side after winding her, she still finds milk there (I can tell from the glugs and the lovely milky moustache afterwards!!)
She's a really quick feeder, and the tendency, having a 3 year old as well, is to think she's finished the first time she spits the nipple out - you know what it's like, there's so much else to do as well. I'm trying really hard though now to persist in winding her and trying her back on the same side.
I think the hardest thing is that the frenzy always seems to happen at the time when your older one needs you the most, and when everyone is really knackered.
I know you don't really want to hear that someone is still having the same experience as you at week 8, but trust me, you will find your own ways of managing both of them.
I distinctly remember being in floods of tears in dd1's bedroom during week 1, looking down at her sleeping peacefully, because I'd missed giving her a goodnight kiss before she fell asleep as I was feeding dd2 in another room (dh had put dd1 to bed). I remember thinking, "I'll never manage to deal with both of them at the same time", and mourning the loss of the time when it was just the 3 of us, and the time when she was as small as dd2. But now we're finding ways of accommodating the needs of both of them. Feeding sitting on the loo seat may not be my best place to feed, but it means I can play bath games with DD1 - I also do this during stories before bed.
One thing that seems to work for me is actually bathing DD2 first, getting her ready for bed and giving her bedtime feed during DD1's bathtime. We often have to break off (not my preference but needs must), but at least I know that whilst I'm brushing teeth, wiping bottom etc., if DD2 cries, it's because she's tired, not because she's hungry. I tend to park DD1 in front of a video after tea, in order to get this done. I also do her hot meal at lunchtime now, and always have a cold sandwich tea, as this can be prepared well in advance if necessary.
Hang on in there with the feeding though - particularly as it's going better for you second time around - you will find your own ways of doing things.
Good luck