If you read "ἄκων" as "unwilling" as most current version do, then that makes this a bit easier to understand what he means.
Paul is not claiming that his preaching is unwilling, but is offering a hypothetical: but if (εἰ). What he is saying, that his will is irrelevant on whether he preaches the Bible or not - he is obliged to since that (preaching the gospel) is his obligation (or stewardship.)
This serves as a contrast to that which he has a choice in, and which is the core thing he speaks of here - Paul's refusal to accept support from the Corinthians.
15 But I have used none of these things. And I have not written these
things so that it will be done so in my case; for it would be better
for me to die than that. No one shall make my boast an empty one! 16
For if I preach the gospel, I have nothing to boast about, for I am
under compulsion; for woe to me if I do not preach the gospel. 17 For
if I do this voluntarily, I have a reward; but if against my will, I
have been entrusted with a commission nonetheless. 18 What, then, is
my reward? That, when I preach the gospel, I may offer the gospel
without charge, so as not to make full use of my right in the gospel.
Paul can boast of not being a burden to the Corinthians, since that is his choice and liberty. In contrast, he is obliged to preach to the gospel: "woe to [him] if [he] do[es] not preach the gospel"