6G is envisioned to empower wireless communication and computation through the digitalization and connectivity of everything, by establishing a digital representation of the real network environment. Mobile edge computing (MEC), as one of the key enabling factors, meets unprecedented challenges during mobile offloading due to the extremely complicated and unpredictable network environment in 6G. The existing works on offloading in MEC mainly ignore the effects of user mobility and the unpredictable MEC environment. In this paper, we present a new vision of Digital Twin Edge Networks (DITEN) where digital twins (DTs) of edge servers estimate edge servers' states and DT of the entire MEC system provides training data for offloading decision. A mobile offloading scheme is proposed in DITEN to minimize the offloading latency under the constraints of accumulated consumed service migration cost during user mobility. The Lyapunov optimization method is leveraged to simplify the long-term migration cost constraint to a multi-objective dynamic optimization problem, which is then solved by <inline-formula><tex-math notation="LaTeX">Actor</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">Critic</tex-math></inline-formula> deep reinforcement learning. Simulations results show that our proposed scheme effectively diminishes the average offloading latency, the offloading failure rate, and the service migration rate, as compared with benchmark schemes, while saving the system cost with DT assistance.