You need an Apple ID to submit a podcast in the first place, but it's not so important once you've been accepted. Access to the feed is accepted as authority to modify it, so you don't have to do anything to add episodes except add them to the feed and re-publish it: the rest follows.
You should make sure that your own email address is in the 'itunes:owner' tag in the top section of the feed: this is what it used should the Store staff need to contact you. Apart from that I don't think the ID really matters; if you are still concerned about it you would have to ask Support about this. Go to https://itunespodcasts.zendesk.com/hc/en-us/requests/new and click the disclosure triangle at the right of the 'Choose a topic' field - choose 'Other' and proceed from there.
You say you want to 'post new podcasts' but I assume you really mean episodes in the same podcast, so as I say you can simply add them to the feed - it doesn't matter what ID you are signed into iTunes with. If you really want to start a new, separate, podcast with its own collection of episodes and a different overall title then you can just sign in with your own ID and submit it - it won't affect the original podcast in any way.