In addition to a tax treaty, Italy also has a "Totalization Agreement" which relates to coordination of social security/social insurance payments. So, in your case, your social security benefits may be taxable by Italy under this arrangement. If so, then the agreement will also allow a credit or exclusion of those benefits on your US tax return.
The rest of your understanding about how the credits work is correct.
Italy generally has higher taxes than the US (though like the US, some types of income are taxed at different rates than others), so it is unlikely that you will have a balance due on the US after credits are applied (and/or any tax treaty claims are made.)
However, there are many variables that come into play so there is no real guarantee. You will want to prepare your Italian tax return before working on your US tax return. This may mean that you will need to file an extension.
To allow time for you to file the foreign return and receive documents in your new home, you are automatically granted an extension to file (but not to pay) of June 15th when you live abroad.
However, you may wish to apply for an additional extension to October 15th to allow additional time to get everything sorted.
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