No. For married couples filing separately, the SALT deduction limit is $5,000 per return. For all other filing statuses, the limit is $10,000.
You will not be able to claim $10,000 each if you decide to file separately.
State and local taxes (SALT) are the taxes on lines 5a (state and local income tax or general sales tax), 5b (state and local real estate taxes) and 5c (state and local personal property tax) of the Schedule A.