I worked full time when I found my house with a private landlord a few years back but moved to part time a year ago and now receive some HB, so I may have this issue if I want to move. However, I'd take exactly the same approach I did last time.
I put a friendly sounding ad on the wanted section of Gumtree and on my local city's forum on the wanted property section, asking for a 2 or 3 bed house in xx area with garden and parking (either drive or on road). I then listed all the positive things about me - that I worked, had paid a mortgage on my own house for 7 years with no missed payments and that I'd be willing to set up a direct debit for the rent. Your positive thing is that you have cash upfront and can offer a guarantor. I also said I was clean, tidy, responsible, houseproud and enjoyed gardening. I also mentioned that I had a DS and gave his age, plus a very old cat that sleeps all day, so the property had to be ok for children and pets. One thing I didn't do was mention a price I was willing to pay up to - I left it open. I think that sometimes if you say something like 'I can pay up to £500 pcm', a landlord that's out to make as much money as possible may then say their property costs £500pcm, even if they were only going to rent it for £450 until they saw your ad.
I had two people contact me off the back of that, both via Gumtree. One had a lovely house but wanted £700pcm, and although they said they'd be willing to accept £600 for the right person, it was too expensive for me. The other was my landlord. He initially wanted to let it for £585, but I negotiated down to £500pcm by comparing it with other rents for similar houses in the area (from Rightmove) and saying I was willing to take a longer contract than the standard 6 months. He's been a fab landlord, and was great when I moved to partial HB, mind you he had no reason for complaint because I ensured the direct debit was still paid even though it took the council 7 weeks to start my HB payments, which left me in loads of debt! But all sorted now.
Also don't give up on agencies - some have landlords that will accept HB. Best bet is to visit the ones in the area you want to live in and explain your situation, pointing out all your positives (see above!), and ask them to contact you when something comes up. Also ring them if there's something you like, because you never know.
Good luck.