ECDSA is advertised to prevent existential forgery even in the face of an adaptive chosen-message attack. That is: even if the adversary can control the messages you sign, not just see a few signatures, they still can't find a single message/signature pair for a message they didn't ask you to sign.
As part of your obligations of the ECDSA security contract, of course, you must choose the per-signature secret $k$ independently and uniformly at random; if you fail to do this—if you repeat the per-signature secret, or if your choice of it is even substantially biased—then you might leak the private key. RFC 6979 defines a way to derive the per-signature secret by a pseudorandom function of the message under the secret key, which obviates the need for an entropy source at signing time—or at least mitigates a broken one.
But you could also just use Ed25519, which has this design built-in so you never need to worry about an entropy source at signing time, and has a clearer security story, and is an all-around better design.