What happens to the old nature's desires when the Holy Spirit is in control?

When the Holy Spirit is in control, the desires of the old Adamic nature are not fulfilled and do not come to the fore. The Spirit's operation restrains the lusts of the flesh, so the believer will not carry out things like stealing, adultery, or gossip.

Les explains that walking in the Spirit means walking in the power and control of the Holy Spirit. In that state, Paul says, you will not fulfill the lusts of the flesh. The Holy Spirit will never tell a believer to steal, commit adultery, or gossip. Although the old Adamic nature still lurks in the background and lusts, or desires, when the Holy Spirit is in control, those things do not come to the fore.

Galatians 5:16

This I say then, Walk in the Spirit, and ye shall not fulfil the lust of the flesh.

Why don't the desires of the old nature show up when you walk in the Spirit?

Walking in the Spirit means living under the Holy Spirit’s power and control. Because the Spirit never leads a believer into sin, the old Adamic nature’s lusts—though still lurking—do not come to the fore or get fulfilled. The Spirit’s control prevents those desires from surfacing and governing behavior.