The 1099-S is more or less irrelevant in terms of reporting the exchange. Closing agents are just required to issue them.
With respect to Form 8824 see:
https://www.irs.gov/instructions/i8824#en_US_2021_publink12597kd0e204
Reporting of multi-asset exchanges.
If you transferred and received (a) more than one group of like-kind properties, or (b) cash or other (not like-kind) property, don't complete lines 12 through 18 of Form 8824. Instead, attach your own statement showing how you figured the realized and recognized gain, and enter the correct amount on lines 19 through 25. Report any recognized gains on your Schedule D; Form 4797, Sales of Business Property; or Form 6252, Installment Sale Income, whichever applies.
