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Southern Baptist Traditionalists prefer to believe that the gospel message is sufficient as "prevenient grace" and that anyone can respond if they have heard the gospel. In the Traditionalist view, because the New Testament (and the Old Testament, for that matter) was inspired by the Holy Spirit, that the Holy Spirit works through the New Testament and the gospel to make humanity's hearts responsive.
Arminians prefer to believe that the Father sends prevenient grace through the Holy Spirit that regenerates ("brings to life a capacity within the heart, wherewith they can respond positively to the gospel") the believer's heart so that they may believe even with total depravity. Arminians typically take a stronger view of total depravity.
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The statement made that established "Traditionalism" as a term for this long-held belief is here, and I believe accurately reflects the views of most (Southern and Independent) Baptists who hold this point of view: http://connect316.net/the-statement/
You are right in that they don't hold the same view of prevenient grace as a Methodist does. Methodists/Arminians actually commonly accept Total Depravity, but many Baptists do not. The other way that Traditionalists would explain things is by agreeing with the necessity of something like "prevenient grace", and that God has given such to everyone, and that it still may be resisted.