I not sure what you mean by this statement " Sharjah taxis cannot pick up in Dubai and vice versa".because I also live in sharjah and just took a taxi home from Dubai last night and since they added the 20 AED extra charge Taxi drives do not mind going to sharjah.I just passed by the Sharjah airport yesterday and right in front of my car was a orange and white public bus going to the sharjah airport,so I would ask at the info desk in the airport to get some info on this.
You could also try to save a little money and go to the Mirdif town center then get a taxi from Mirdif to Dubai,which might save you the extra 20 to Dubai,so ask the taxi driver how much the full cost would be to go to Mirdif and compare.
One of the best sigth seeing things in Sharjah is to the left of the airport The Breeding Centre for Endangered Arabian Wildlife which is just right down the highway very very close so look on their website.There is also a new aquarium in sharjah which I also hear is very nice.
Unless you have your heart set on Dira city center,I would past that up and either go to festival city [mall] which is easier to get to then DCC since you can stay on the 311 emirates highway and skip the traffic going to DCC or go to the Dubai Mall where you can do the same traffic wise and its huge,it has its own aquarium and musical fountain. .