You have a few choices; I would say that St Vlas, Ravda or Nessebar would suit what you are looking for. These are all close enough to Sunny Beach but at the same time, far enough away that they still retain a little bit of what Bulgaria is really like.
You have the choice in all 3 of these, to either rent an apartment or book into a hotel yourself and it usually works out cheaper. We've rented apartments in St Vlas and Ravda through holidaylettings.co.uk and villarenters.com in the past.
Heading to this end of the country allows you to book low cost flights through wizzair.com, they fly direct from London Luton to Bourgas throughout the week so you've plenty of flights to choose from. If you go for one of the tour operators you're more or less restricted to 7 or 14 day holidays.
There is a couple who moved over to Bulgaria a few years ago, who run pickups from the airport to apartments etc their email address is keiamand@yahoo.co.uk. If I can remember correctly they charge you 60 levs either way and compared with taxis it's a lot cheaper.
If you're looking for some ideas of what Bulgaria's like, have a look at lonely planet's website, they give you loads of info and obviously us guys on here love Bulgaria so we'll keep you right.
Give us a shout if you want to know anything else