Firstly, most of the caves in the Dales have vertical pitches in them. Did your previous caving trips include SRT? I'm afraid your climbing gear isn't appropriate for SRT: e.g. ropes need to be "static" (low stretch), the harness has a different fit and needs a chest strap and an attachment point for your ascenders, descender and cowstails.
Secondly, why alone? There are a lot of caving clubs in the Dales which run weekend trips of all grades. If you can't make weekends, once you join one and get to know people you will probably find that there are mid-week trips arranged by those who are retired, work shifts, etc.
If you aren't experienced at SRT: how to rig safely and then e.g. how to pass hanging rebelays, you absolutely must have training. Clubs will do this, as will the CNCC, or there are several companies (e.g. Yorkshire Dales Guides) who can do it.
Going solo, at the very minimum:
- Leave a callout with a responsible person giving your route (some systems have many different passages in them) and expected time out.
- You'll need two ascenders and a descender (a figure of 8 isn't safe for re-belays, since you have to detach it from your harness), cowstails, etc.
- Static caving rope. Caving clubs have their own ropes for organised trips.
- A headtorch and a backup (especially important for solo trips)
So, in summary, join a club. Certainly the one I'm in (the BPC) and I do not doubt all the others have members of all ages and are very friendly and welcoming to new members.