For me the pick would be Oak Flats. Its east of the main road (Pricess Hwy) closer to the lake and is mainly older houses. Be careful the are some industrial areas and also some busy roads used to transit through the area. Oak Flats is close to rail and freeways to the north and south and is services by the local buses. As you state its close to Shellharbour Square (Stocklands not Centro) which is about to undergo a major revamp. The hospital although major to the area is relativtely small so to me not a real draw card. These areas are all about 25min to the CBD of Wollongng by road.
To me Warrilla, Barrack Heights, Lake Illawarra, Mt Warrigal are all similar and contain alot of NSW Housing and ex Govt houses and there are areas there that definitely have issues relating to anti-social behaviour etc. Saying that there are also nice street in all these areas but no for me i'm afraid but probably the only areas in your price range.
The vacancy rates are low in all of wollongong/shellharbour area with most places that are competively priced being snapped up quickly. Mates house went in 3 days.
The only drawback with some of the areas like Ablion Park is that Captial Gain has been relatvely low with many houses we looked at having been sold previously in the last 4-5 years for not much more that the asking price today. For example this house was PI at auction for $365K offer upto $370K post auction but rejected by vendor but now on listed for sale at $370K. needs plaster work to some ceiling and painting etc but sold in 2007 for the same price. Saw many like this in our search.
http://www.domain.com.au/Property/For-Sale/House/NSW/Albion-Park/?adid=2008622297
The best idea would be to spend the time and have a drive around to get the feel for the areas.
If you have addresses I can provide you some specific feedback with a drive by as I am in the area every day.
Just for perspective, I own and live in my ex Housing house in a not so nice area of Wollongong (Unanderra=Struggletown, had Polair 100m above my house a few weeks ago and always makes the headlines in the paper everyweek but thats another story) and own a $400K IP in the Shoalhaven so have no axe to grind in my assesment above.