If it's an organisation account, then really the only things you might reasonably benefit from following are those of partner organisations that you might expect to retweet. Eg. The campaigns account of $cycling_organisation might occasionally retweet announcements from the social rides account, or the council's invitations to respond to relevant consultations.
Organisation accounts tweeting or retweeting about irrelevant stuff is a good way to get unfollowed. Less is more, as you've just discovered.
Follow things of general interest on your personal twitter account.
Historically, Twitter required people to be following each other in order to have a conversation in Direct Messages, which might explain why so many accounts are being followed. AIUI there's now a setting organisations can use to allow them to receive DMs from anyone.
I don't know about Mac specific software, but you can use
Tweetdeck on the web, which is probably helpful for high-volume stuff. Bear in mind that Twitter have been working to kill off third-party clients by repeatedly changing and restricting the API, because usability interferes with the important business of selling users to advertisers.