No, "follow me" does not mean to follow after OC law (which law was done away with for those in Christ), but to follow after Jesus specifically as the Savior, not OC law. Those who seek to establish their righteousness by the keeping of it are under God's wrath; those He saves are under grace. By informing us that He would fulfill the law, He informed us that “we don’t have to” follow law for righteousness - righteousness is imputed solely from Christ, by Christ, through grace. There is no righteousness to be found by keeping the OC law, and so we are not governed by it.
For those to whom it is given, salvation comes as a completely free gift from God; as such, our obedience provides no part in the receiving of it. For those saved, the Old Covenant law is done away with in Christ. I'll post the same verses I posted to SabbathBlessing:
[2Co 3:10-18 KJV]
10 For even that which was made glorious had no glory in this respect, by reason of the glory that excelleth.
11 For if that which is done away [was] glorious, much more that which remaineth [is] glorious.
12 Seeing then that we have such hope, we use great plainness of speech:
13 And not as Moses, [which] put a vail over his face, that the children of Israel could not stedfastly look to the end of that which is abolished:
14 But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which [vail] is done away in Christ.
15 But even unto this day, when Moses is read, the vail is upon their heart.
16 Nevertheless when it shall turn to the Lord, the vail shall be taken away.
17 Now the Lord is that Spirit: and where the Spirit of the Lord [is], there [is] liberty.
18 But we all, with open face beholding as in a glass the glory of the Lord, are changed into the same image from glory to glory, [even] as by the Spirit of the Lord.
Correct- Salvation is given as completely free gift from an exceedingly gracious, kind, loving and merciful Father through Jesus Christ the Savior to those whom He had chosen to salvation, even though none deserve it. To whom it is given, it is given in its completeness, with nothing withheld or dependent upon our satisfying of any stipulations or requirements, besides of having been chosen by Him for it.
No, He did not fulfill the law to show what true obedience looks like that we should be required to do likewise. He fulfilled the law so that His elect wouldn’t have to so in order to become saved. By Christ, for those He saves, OC law has been abolished, and salvation is only by His grace not law.
True obedience is to have faith in Christ alone as the Savior, not to the law. Obedience also comes as a gift from God unto those that He saves.
[Rom 1:5 KJV] 5 By whom we have received grace and apostleship, for obedience to the faith among all nations, for his name: