Do my monthly payments for medical/dental insurance through my employment count as health coverage tax credit?

You can only deduct them if they come out of your checks after tax, not as a pre-tax Deduction.  Ask your employer if you don't know.

But it may not be worth it to Itemize.  You can only deduct the amount of unreimbursed Medical Expenses you actually paid over 7.5% of your AGI.  And then all your itemized deductions have to be more than the standard deduction to get any benefit (so you would only be getting the benefit of the amount that puts you over the standard deduction).  And since the Standard Deduction is increased more people will not need to Itemize.

For 2020 the standard deduction amounts are:

Single 12,400 + 1,650 for 65 and over or blind (14,050)

HOH 18,650 + 1,650 for 65 and over or blind

Joint 24,800 + 1,300 for each 65 and over or blind

Married filing Separate 12,400 + 1,300 for 65 and over or blind