Persevering rhymes with:
revering, veering, adhering, appearing, auctioneering, bioengineering, careering, cheering, clearing, dearing, deering, dering, disappearing, domineering, earring, electioneering, engineering, fearing, gearing, gering, imagineering, interfering, nearing, oceaneering, pioneering, premiering, profiteering, racketeering, rearing, reengineering, revering, schwering, searing, shearing, smearing, sneering, spearing, steering, veering, volunteering
Persevering sounds like:
perseverance, perspiring, prospering