Have you been in to see the housing officer in person?
They can offer assistance with:
Negotiating with your landlord or more time
Providing a deposit and helping with guarantors for private letting a
Putting you on the housing list - this usually a bidding process an you should already be on it as you've been served notice.
If you are not on the housing list already, get the paperwork done ASAP and they can increase your banding when you are actually homeless.
By being on it, you should be able to bid on a range of properties in your area (broad radius). You can see immediately where you are placed on the bidding list, or where you would be if you put in a bid. If you're in the top ten, then you have a reasonable chance as there's a secondary grading of the top bidders on a property which may mean you'll be lucky.
Also, phone shelter, visit the CAB and keep phoning the housing office for updates.
This exact thing happened to me and I was told the same as you In the first instance, but I kept pushing and found it wasn't as dramatically awful as they made it sound. Essentially, they want you to fix your housing issue by yourself and horrible as it sounds some housing officers will fob you off if they think you have a chance of getting yourself sorted without their help.
I got my house after just 5 weeks of bidding (before I'd been evicted) and though I had to move 25 miles and get a new job etc., having housing security massively helped.
Best of luck. It's very scary and it made me quite ill with worry ( the last thing I needed as a single parent with a six month old!), but it really did work out in the end.