I agree with you that the condition that we are delivered/freed/rescued FROM refers to more than his body. I will argue that in Rom 7:24 the condition that Paul meant is what NLT translates as "life that is dominated by sin and death". I'm using Craig Keener's 2011 book Romans: A New Covenant Commentary (NCCS), especially his excellent Excursus: The "Flesh" (sarx) in Romans.
First, some terminologies:
We need to properly interpret the meaning of an extremely important concept critical to understanding Paul: sarx, frequently translated as "flesh" (ESV) or "sinful nature" (NIV & NLT), which is used in the next verse: Rom 7:25.
Then we need to connect it with sōma as in "sōmatos tou thanatou" (Rom 7:24) and "sōma tēs hamartias" (Rom 6:6), frequently translated as "body" (ESV & NIV) or "life" (NLT).
We also need to connect the Greek sarx with the Hebrew basar that
the OT employs for humans in their limited creatureliness, including their mortality.³⁵ By Paul's day, some Jews employed the term for human weakness in its susceptibility to sin.³⁶ Basar and its Greek translation sarx were not inherently evil, but as "human weakness" were susceptible to sin. Flesh was not meant to lead human life, but to be the arena in which life should be lived in obedience to God. (emphasis mine)
Second, Rom 8:2 tells us the agent which delivered/freed/rescued us: the "power of the life-giving Spirit" (NLT) or "the law of the Spirit of life" (ESV).
With these terminologies squared away we are one step closer to answer the question (i.e. what condition we are delivered FROM). The answer: humanity as sarx, which I believe is captured better by the NLT translation: "life that is dominated by sin and death" as contrasted with the "life controlled by the Spirit" (Rom 8:9). I believe this reading is supported by Craig Keener's analysis of the meaning of sarx and sōma in the following quote from his commentary:
Excursus: The “Flesh” (sarx) in Romans
Neoplatonic and gnostic dualism absorbed by later Christianity denied that the body was good³², and many scholars today, reacting against this conception, argue that Paul’s use of sarx bears little relation to sōma, “body.” Some translations (such as the NIV) even poorly translate sarx simply as “sinful nature” (which for some Christian traditions also evokes a dualism of two natures struggling within the believer).
Paul certainly believes that the body can be used for good (12:1)—but also for evil (6:13). The body as such is not evil, but if the body’s desires rather than God’s Spirit dominates one’s existence, one readily comes into the power of sin³³. Paul speaks of the “passions” (1:24; 6:12; 13:14) and “desiring” what is not one’s own (7:7–8; 13:9). Impure hearts lead to defiling bodies sexually (1:24); the old life in Adam involves the “body of sin” (6:6); one should avoid obeying the desires of the mortal body (6:12); the existence of moral defeat is characterized by the “body of death” (7:24; cf. 8:10–13).
Paul connects the term “flesh” with the body. Ultimately this is not an anthropological dualism between two parts within a person (in 7:23, the mind remains subject to the flesh), but humanity as sarx contrasted with God’s Spirit (8:4–9; Gen 6:3).³⁴ The OT employed the equivalent Hebrew term basar for humans (or other animals) in their limited creatureliness, including their mortality.³⁵ By Paul’s day, some Jews employed the term for human weakness in its susceptibility to sin.³⁶ Basar and its Greek translation sarx were not inherently evil, but as “human weakness” were susceptible to sin. Flesh was not meant to lead human life, but to be the arena in which life should be lived in obedience to God. Paul could use “flesh” also for the outward existence (Rom 1:3; 2:28; 4:1; 9:3, 5, 8; 11:14), again what is merely human (though not intrinsically evil) rather than empowered by God (he sometimes contrasts flesh with the Spirit or the promise).
Despite these observations, flesh had an inescapable bodily dimension.³⁷ Contrary to the views of a large proportion of NT scholars, mostly following secondary sources, Diaspora Jews by Paul’s day commonly did distinguish soul and body, often expecting immortality for the former even when affirming resurrection for the latter.³⁸ In 7:5, sinful passions working in the body’s members characterize being “in the flesh.” In 8:13 one is either destined for death in the flesh or resurrection by putting to death the body’s deeds. Flesh is also linked with the body in 6:19; it contrasts with the mind in 7:25; the law working in one’s bodily members (7:23) is the law of the flesh (7:25).
The conflict between the law of sin in the members and the law in the mind in 7:25 was not the basis for the verdict of “no condemnation” in 8:1, as if God would overlook physical sin provided one’s mind harbored good motives. Far from it: 8:1–13 contrasts those who serve the flesh with those who serve God by the Spirit! Paul’s goal is a way of thinking dominated not by the flesh (hence by one’s physical desires, which have a legitimate place, but not in ruling life), but a way of thinking dominated by the Spirit (8:5–9). This new way of thinking involves a renewed mind (12:2). This renewed mind teaches believers how to present their bodies in the service of the larger body—the body of Christ (12:1, 4–5). Such a mind is no longer self-centered, but Christ-centered; no longer seeking full autonomy, it now submits gladly to the greater good of God’s purposes. “Flesh” is the localized self in contrast both to dependence on God (through the Spirit) and the corporate interests of Christ’s body. Life ruled by the flesh is, at root, human selfishness and self-centeredness (or sometimes centered in one’s group), rather than genuinely altruistically sharing God’s interests. Paul’s goal is not to annihilate self, as in some religions, but to connect it to the service of a greater purpose (cf. 12:1–8; Matt 6:33). Paul was no gnostic, but neither was he a hedonistic Western individualist who keeps religion in its subjective place.