The 627 will handle copious amounts of .357 Magnum ammo without issue and there is no reason to restrict your usage of the gun to .38 Special ammo if you don't want to.
The "shoot .38's and carry .357's" advice is usually applied to K-Frame Magnums including the models 13, 19, 65, and 66. Even with those, that advice is somewhat exaggerated as even K-Frames can typically withstand thousands of rounds of full power .357 Magnum ammo so long as bullets no lighter than 140gr are used. The weakness of the K-Frames was that the forcing cone was thinner in the six o'clock position and somewhat prone to cracking in that spot if fed a steady diet of lightweght magnums (the full power 125gr loadings were the most notorious offenders).
The N-Frames such as your 627, however, have very thick forcing cones and I don't believe I've ever even heard of one cracking much less seen it in person. Whoever told you that your revolver was not up to a steady diet of .357 Magnum ammo was severely misinformed.