I’m going to assume that RSA has the security properties that we hope it does. In particular, I’ll assume that, if someone knows a message p, a public key pubKey, and the value of enc(p,pubKey), then it will be hard for them to calculate the corresponding private key, priKey. And, even if they have lots and lots of plain text / cipher text pairs, your private key still will be hard to compute. Hard to compute here means they’d either need a lot of time or a lot of hardware or both. (There is no proof of this assumption, but it seems to be true that know one knows of a way to break RSA when enough bits are used.)
Suppose you encrypt (rather than sign) a message m with your public key, and send someone the result enc(m, pubKey). And suppose that they know your public key too. From the assumption in the first paragraph, with m in the role of p, your private key will be safe. And even if they somehow know m, it will still be safe.
Signing usually means encrypting a hash of your message with your private key, i.e., you send (m, s) where s = enc(h(m), privKey). Then the recipient with your public key can check that dec(s, pubKey) = h(m), which means that s is enc(h(m), privKey) and so (almost certainly) must have been computed by someone who knows privKey.
(This chain of inference in the previous paragraph actually relies on a property of RSA and h that wasn’t stated in the first paragraph, namely that, if someone knows pubKey, but not privKey, it is hard for them to compute a pair (m,s) such that dec(s, pubKey)=h(m). But this is completely irrelevant because the question is about signing with the public key.)
If you mistakenly sign with your public key, then you send (m, enc(h(m), pubKey)). Now the recipient or eavesdropper will know h(m), enc(h(m), pubKey) and, presumably, pubKey. This is the same as the situation described in the first paragraph with h(m) playing the role of p. So again your private key is safe. (The recipient also knows m, but knowing m shouldn’t be of any help unless the message says something like “my private key is ... .“)