Can I use a 60A circuit for both a cooktop and oven?
That 8/3's overbreakered no matter what
The 8AWG copper you have in your walls is rated for 40A at 60°C or 50A at 75°C as per Table 310.16, so you'll have to swap that 60A breaker out for a 40A (NM cable) or 50A (armored cable, SE cable, wires in conduit) no matter what else you do.
40A is adequate for most ranges though
However, due to the Table 220.55 range load demand factors, which also apply to a separate cooktop and oven on a single circuit, 40A is adequate for the vast majority of range setups, and even those that don't fit on 40A have no trouble with a 50A circuit, unless you have a very large cooktop.  In particular, a 12kW nameplate range load only is treated as 8kW of actual load for circuit sizing purposes, and every additional kW or major fraction thereof adds 5% to that 8kW.  So, knowing that a 40A circuit gives us 9.6kW and that ranges are not treated as continuous loads by the NEC, we get 10% over 12kW, or a 13.2kW maximum range load on a 40A circuit.  Going up to 50A gives us 12kW, which translates into a 14.4kW maximum range load.