Large language model (LLM) inference is often constrained by both computation and memory, especially in offloading-based deployments where model weights are transferred across memory hierarchies during autoregressive decoding. In this setting, reducing the number of executed layers can lower token-generation latency while also avoiding costly weight movement. Motivated by this observation, we present **FlexEE**, an early exiting framework for resource-constrained and offloading-based LLM inference. FlexEE makes early exiting practical for LLM decoding through layer-wise exit supervision for reliable intermediate-layer prediction, self-speculative decoding over a Top-K local vocabulary for low-cost exit decisions, and dynamic hidden state management for KV-cache-correct and memory-aware execution. Across generative and downstream tasks, FlexEE enables efficient early exit with minimal accuracy degradation, delivering up to 1.27×/3.16× and 1.25×/2.83× end-to-end speedups on Llama2-7B and Llama3-8B under 0%/50% weight offloading, respectively.