Toilets, Sinks, Showers, Tubs and Disposals - How to remove handles to fix leaky faucet

I want to stop my bathroom faucet from dripping but cannot see how to remove the hot and cold handles to get to the washers (assuming there are washers). I see no brand name on the faucet - it appears to be very generic.

I can tell you that
a) the faucet appears to be something less than new
b) there are no caps or tops on the faucet handles that 'pop off' to reveal screws to remove
c) the faucet appears to be of the inexpensive variety

I can get a flat-head screwdriver under each handle. If I pry and pray, will those handles just pop off of some unseen stem, revealing a screw assembly, or are faucets not designed that way?

Can you offer some guidance concerning how I might get those handles off to let me dig down to the probable leaky washers (what tool(s) should I use?), or do you have another solution I should consider?

Thank you for your time.
Curtis

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