I am trying to add my podcast to Apple Podcasts

Logging into Podcasts Connect to submit a new podcast has been a problem for many people: indeed almost every post from people who are creating podcasts is about this rather than issue with their own podcast. As it's internal to Apple I can only offer the following standard advice, which may or may not help; there really isn't anything else I can suggest.

To start with, you do have to have an Apple ID and an Apple Music or iTunes (on windows) account.

It's become apparent that you cannot log into Podcasts Connect with a Developer ID - see below. You must have a payment method registered in Apple Music/iTunes: make sure your computer is authorised to play purchases (Apple Music or iTunes menu Account>Authorisations>Authorise this Computer). Then try logging in at https://podcastsconnect.apple.com.  One poster found that going into 'private browsing' worked.

For help with the process of creating a podcasting account please see Create Apple Podcasts Connect account - Apple Podcasts for Creators

If this fails I would suggest reporting it Podcasts Support: https://podcastsconnect.apple.com/contact-us , though if you haven't been able to complete the submission you may not be able to access them: in which case the next step is to contact Apple Support: https://getsupport.apple.com

Don't present this as a podcasting problem - they don't handle those and will refer you to Podcasts Support. Present the problem as being the inability to access Podcasts Support. It's also been suggested that getting them to look at your Apple ID may clear matters.

You may also find Robert Walch's post here useful: can't add my podcast to podcast connect -… - Apple Community

As to Developer IDs, Brian Ogilvie has discovered the hard way that you cannot use a Developer ID to sign into Podcasts Connect, and details the steps needed to create a new ID in this useful post.