I built this tool for myself for this same exact problem https://github.com/mraza007/echovault

Anthropic is pushing MCP for a reason. The idea is to spin up a local MCP server that serves project context - docs, architectural decisions, DB schemas - which Claude Desktop and Cursor (via plugins or natively soon) can connect to

For a team this is a total killer feature: you spin up a shared MCP server with documentation, and all agents instantly sync their "brains"

If you are working by yourself, no need to over complicate it, I have three terminal sessions open - Claude, Codex and one for testing that has my AWS credentials.

I tell both Claude Code and Codex to keep the same markdown file updated where I keep my progress, requirements, decisions, thought process etc updated

I use markdown files which full detail and summarized versions. I also create files just for an aspect of it (security, performance) so that it makes it easier to reference.

I saw before that someone has developed a software that can transform the context between major AI programs, which should be helpful to your problem.