Radhe Rahde
I would like to link you to a list of videos for shivling upasana. You said you are a lingayat so I am unsure if you wanted upasana vidhi in your tradition, if not, then these will help you.
First by Pujya Puri Shankaracharya on keeping shivling on home- https://www.youtube.com/watch?v=-tmok_SXUvc
Second by Shri Rajendradasji Maharaj on types of shivling and how shivling should be kept/look & types of materials should be used for shivling (also watch full there is exception for narmadeshwar)- https://www.youtube.com/watch?v=RFx-hbMC7SM
Third: Pooja Vidhi & can women worship Shivling (watch full it tells you how everyone can do pooja and how) - https://www.youtube.com/watch?v=fN_TfYlbJ6Y
Fourth: more on how to keep shivling and do pooja - https://www.youtube.com/watch?v=wQq2Ct9nzJI
Of course, the most important thing is devotion to god, but remember that Narmadeshwar and Pran Pratishtha Shivlings are Bhagawan Sada-Shiv himself, so there are rules since there is extreme power.
If you cannot follow the rules and regulations then don't just keep the shivling lying around or throw it, give it to your local temple with authentic Bhramins so they can do pooja (maybe even in your name).
Also, I would suggest you visit your local temple to do visarjan (unsure have yet to ask Gurudev but maybe they can keep your shivling there, the bhramins will know) or read mantra and do visarjan for you.
Radhe Radhe,
Forgive me if something I said was wrong or the answer did not satisfy you.