I believe you are correct to assume vested shares are yours, or fully vested would imply they are yours regardless of employment with the firm.
Drip shrares are those reinvested so you get additional shares in lieu of the usual cash dividends. Normally you need to opt into this, but it’s possible that this was set as default by the company scheme.
You should have a statement showing exactly the number of each class of shares you own, it’s not overly clear from what you’ve said.
Definitely speak to someone at the company to go through this all. As someone above said, you can end up seriously reliant on an employer if they are paying your wages, sorting out your pension and running a company share scheme with significant value.
like you suggest, if the share incentives are a good deal then keep taking them, but consider spreading things out once the shares are freely yours.