Good first post, and welcome to the Downs!
I think the primary reason Frodo was more deeply affected by the Ring was the relative difference in the power of Sauron throughout most of Bilbo's possession of it as compared to when Frodo inherited it.
Bilbo found the Ring in S.A. 2941. In 2951 Sauron "declared himself openly", and "gathered power in Mordor" (dates from The Tale of Years).
By 3019, the year in which the Ring was destroyed, Sauron's power was at its zenith (sans the Ring, of course). Also, Frodo steadily traveled closer and closer to Sauron in Mordor, and the place where the Ring's power would also be at its, highest, Mt. Doom.
Bilbo never experienced the Ring under such conditions, and even so, as you note, he was barely able to let go of it.
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Music alone proves the existence of God.
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