Thanks - do you have a link to the club you are talking about?
I think
this is the website
As a female cyclist who is increasingly getting into cycle touring, I am lucky to have a husband who wants to/is able to

(he might read this

) come along. Thinking hypothetically here, if he couldn't or didn't want to come along, I think I would still go by myself or with friend(s) ...
At the moment we mostly travel lightly (no tent/sleeping bags/cooking stuff) and stop over at B&Bs, so I can't speak from personal experience as a lone female cycle/camping
I've done lone female hillwalking/wild camping in the Scottish Highlands and thought nothing of it, apart from being very much aware that I MIGHT go over on an ankle or knock myself out, so was watching my steps and not doing anything obviously silly and left rough plans with friends (in the days before ubiquitous mobile phones

). I've also done lots of lone-female hillwalking and stayed at hostels and/or bunk houses, nice places to dry out after a rainy day, sit in the warmth, and have a chat with other "inmates" if I wanted to. Never had any dodgy situations.
Translating that into cycling, I personally would choose any wild camping sites a bit more carefully if by myself (not really much of an option in England anyway, due to legal differences). But "official" camp sites are not normally wild and lawless places that would make me worry about my personal safety, as very often the other campers are families or like-minded people, and you can normally make sure you don't pitch up next to a group of youths out for a big party (which the rest of the campers wouldn't take too kindly to!)
Good luck with the bike decisions! I can see your dilemma with your current Trek bike. You could always go for a couple of short weekend "test" tours, travelling lightly. That way you can then base your decision for a new bike on experience of what worked/didn't work for you
T