I’m using IMAP with Apple Mail.
This summer, my elderly Mac’s storage became choked solid - even though all my data files are on a separate drive.
So a bought a second Mac with a ludicrously large drive, and set up the same email accounts.
Over the next couple of days, twenty years’ worth of emails rattled onto the new Mac via IMAP, all fully searchable etc.
Now I just need to make sure they’re saved as emails on the new machine.
> I use an Email Client, and /think/ I have them all on my computer,
> but I’m not certain.
It depends on how your email client is configured.
If you use the IMAP protocol to connect your client to your email service, then your Inbox is on their servers. Other folders can be created either on their servers on your computer. You can manually copy or move messages between folders (server and local) or set up filters to do that.
Additionally, for your Inbox (and any server folders) you can configure your client to keep a local copy of the folder, for use if/when you have no internet connection. This can be convenient while traveling, but it does cost storage space and some extra transmission & compute time to keep the local copy up to date.
If you use the POP protocol [shudder] your client would normally be configured to download messages to a local Inbox and delete them from the server when downloaded. Consider switching to IMAP to get a richer experience, including the abilities below.
> I don’t mind changing from YahooMail to another email provider, but I
> would like advice on how to Save all my old yahoo messages in
> searchable form.
An IMAP client makes a good tool, both for copying messages from your old service to your new one and for searching your messages.
I use Thunderbird to move messages directly between my personal Gmail address and my work email (on Dreamhost) - Thunderbird has the folders of both open and it is as simple as dragging a message from one service's inbox to the other.
I may be spoiled by my use of Eudora and Thunderbird, but I expect an email client to have very facile search capabilities built-in. Far better than having the messages in some other format and trying to use grep, finder, explorer, or other generic search tool.