Circular soft magnetic dots are the main elements of many proposed novel spintronics devices, capable of fascinating spin-based electronics applications, from extremely sensitive magnetic field sensors, to current-tunable microwave vortex oscillators. Here, we investigate static and broadband dynamic magnetization responses of vertically coupled Permalloy (Py) magnetic dots in the vortex state in layered nanopillars (experiment and simulations), which were explored as a function of in-plane magnetic field and interlayer separation. Under reduction of magnetic field from saturation for the field range just above vortex-vortex ground state. We observe a metastable double vortex state for each of the dots. In this state, novel kinds of spin waves (Winter's magnons along domain walls between vortex cores and half-edge antivortex) are excited. For dipolarly coupled circular Py(25 nm)/Cu(20 nm)/Py(25 nm) trilayer nanopilars of diameter 600 nm, a small in-plane field splits the eigenfrequencies of azimuthal spin wave modes inducing an abrupt transition between acoustic (in-phase) and optic (out-of-phase) kinds of the low-lying coupled spin wave modes. Qualitatively similar changes (although more gradual and at higher values of in-plane fields) occur in the exchange coupled Py(25 nm)/Cu(1 nm)/Py(25 nm) trilayer nanopillars. These findings are in qualitative agreement with micromagnetic dynamic simulations.