The way I normally upload audio files (or any large file) is to use an FTP client like FileZilla (if you download FileZilla, you want to download the Client, not the Server software). What you will need from your host is an FTP user ID, password, and the IP address to the server where your site is located. Once you have that information, you use FileZilla to connect to your site, and the interface is a lot like Windows Explorer. A representation of your local computer will be on the left pane, your site will be on the right pane, and you can drag & drop files across from one side to the other. You can use FileZilla to create subfolders, too (by right-clicking on a folder). So what I would do is create a folder called /recordings, upload your audio files to that folder, then in your post or page, add the HTML or shortcode that points to that file, like:
Make sure that you are using the Text Editor instead of the Visual Editor when entering HTML or shortcode. Also make sure there are no spaces in the file name, because something seems to go wrong with the audio shortcode when there are spaces in the file name.
Instead of an FTP client, you can also check to see if your host has some sort of control panel interface to your site. Most of the larger web hosts will have something that you can use to create a subfolder and upload files.
Either a control panel or an FTP client should allow you to upload files of virtually any size, so you can bypass the file size limits of WordPress.