Font licencing with closed Adobe CC subscription

All that happens after you cancel your subscription and sell the logo to the client is that you can no longer edit type in files that use that font until you subscribe again or purchase a new font license.

You should be able to subscribe to an Adobe plan at a higher monthly price than an annual plan to be able to cancel at any time (effectively having a one-month subscription).  I'd call the purchase by phone number at 800-585-0774 as the web portal seems slanted toward the annual plans.  This will allow you to install the font and edit files that use it.  

However, you may be able to purchase a perpetual desktop license for the fonts you would like to use instead.  If it's an Adobe-owned font, there are three resellers to check with.  If it's a third-party font it can probably be purchased from the font foundry’s website or their authorized reseller.

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Here is the font licensing for Adobe fonts.

Scroll down to "Can I keep using files I created with these fonts if I cancel Creative Cloud?" and "Are these fonts available with a perpetual desktop license? What if I need a custom license?" for the complete answers to your questions as well as links to resellers for Adobe-owned fonts.

Generally speaking, I never leave any lettering in logos, wordmarks or other kinds of logo-like graphics as live font-based objects. I convert the lettering to outlines when I'm finished editing. In the short term it makes it easier to share the logo with clients or other people authorized to handle the asset. Over the long term the logo file will open in applications like Illustrator more reliably. I've seen graphics files more than 20 years old containing live text objects do some unexpected things when re-opened. The way fonts get handled or the way type-based effects get rendered can change over time. Then of course some font formats wind up no longer supported. 20+ years ago Illustrator was a Postscript-based application that included 200 or so Postscript Type 1 fonts. Today Postscript fonts are no longer supported in Adobe's applications.

If you currently have an active Creative Cloud account and are looking to close it I would recommend converting the type objects you have in a logo design to outlines. It's either that or you'll have to look at buying a license of the same typeface (unless it's something that is freely available, such as fonts from the Google Fonts site).