For electrical items try Carrefour, Hyperpanda, DG Sharaf or Plug-ins.
For other furniture look at all the general home stores in the malls, as well as IKEA, plus less well known places like Safita or Pinky's that have real bargains for nice items. No need to travel to Sharjah if you know where to go in Dubai. Also look on supermarket notice boards, on Dubai Property Real Estate for Sale and Rent. Jobs in Dubai, Classifieds in Dubai with Dubizzle.com and on our classifieds for secon hand items for sale. As there is a fairly transient population there are always nearly new items going for a good price.