Yes - for the remaining c. 5.5 years of your FTB relief period. And it's not dependent on you selling your former home. I am assuming that you will occupy the new property as your Principal Private Residence.
And you can only ever have a single PPR but having additional properties (e.g. one or more rental properties) does not preclude you from claiming FTB relief - unless you had previously bought a property before buying your first PPR.
Bear in mind that if you rent the old PPR out within 5 years of purchase then you will be liable for a clawback of stamp duty. And if you keep it for more than 12 months before selling it on then some portion of any resale gain will be assessable for CGT.
It won't unless you don't sell it for more than 12 months after vacating it as your PPR. Then some portion of any gain will be assessable for CGT. For example if you owned the property for x years and then hold onto it for y more and then sell it then (y-1)/x of any resale gain will be assessable for CGT.