Teaching jobs are plenty. There are AD is the Age & the Heraldsun for Melbourne. The SMH for Sydney. Most jobs are contract few years then review.
Your engineer job is not really promising at this time in economic cycle. Qantas has just reduced its workforce and all big R&D companies had do the same in the last 2 years . If you have a job now, then if I were you I stay in UK unless you are a really social networker or know someone can help you pass the door..
Living cost for both would easily be $1800-2000 eg food, bills, insurance, accommodation 1 bedroom, eating out once a week and No car. Food will be way way cheaper compare to UK. Local Restaurants are plenty and much cheaper. Melbourne and Sydney are beautiful. Sydney rains too much, bad traffic congestion but very wonderful in Summer. Melbourne is more livable.
Our family has Nando meal in UK cost us $68 compare $37 in Melbourne.