If job is in Sharjah, personally, I would live in Sharjah then.
The travel between the 2 emirates can be horrendous, and Sharjah has cheaper housing (your 108,000dhs per year housing would go a lot further...in Dubai...maybe a studio apartment, possibly a 1 bed)

Are you aware that it is illegal to live together here if you are not married.
Sure, there are some that do it, but you need to be careful.
You will not be able to get a lease under both names, nor will you be able to sponsor her.

The salary should be OK for 1.
Do you need to get a car?
Please factor in car/petrol, food, utilities etc