We spend $60B/month more than we take in and must fund that $60B gap with additional debt issuance, which we can't do unless we increase the debt limit. The only way to avoid increasing the debt limit is to reduce expenses by $60B/month. Prioritizing debt principal and interest payments doesn't avoid default if you cannot issue new debt because you've hit the debt ceiling ... and you must issue new debt to both replace existing debt falling due and fund the additional $60B/month ... unless you cut expenses by $60B/month. There is no way around it. Now there are half a dozen short-term schemes being discussed to avoid the grim reaper for a month or two, but they are either legally questionable or just won't work: 14th amendment, issuing high-coupon debt, etc. and even then it's only a short-term stop-gap and we're back to the wall in December.
There is so much confusion surrounding this issue, particularly conflating non-funding (CR) and the debt ceiling.
Bottom line: unless Congress increases the debt limit, the US will default on its obligations at some point, like a few weeks, after October 17th.