@fsmithjack, may I ask how many LP's you own? I ask because if you gave up on a table/arm/cartridge (twice), I'm guessing not many. Is acquiring a large LP collection an ambition of yours? An LP player and an LP collection go hand-in-hand.
You will need not just a good LP player, but a good record cleaner (at least a vacuum system, if not an ultrasonic), a willingness to clean the stylus between LP sides, and use of an LP cleaning brush of some type with every play (for dust removal). The LP is a demanding mistress! And not cheap; new LP's are almost always twice the price of the same album on CD. Going with LP's takes dedication, and attention to detail. A Vinyl Flat is also a good idea; many used LP's (a great source of a lot of music) have become warped from poor storage by ignorant former owners (not to mention their grooves damaged by poor handling and a cheap player) , and a fair percentage of new LP's are poorly manufactured. Be prepared to have to return LP's for non-defective copies.
Those who grew up with the LP as the only serious source of music learned the LP ropes little-by-little, the knowledge, collection, and player growing together. Learning about LP's (including different pressings of the same album) is fun and rewarding, but not carefree. If you want your third stab to be more satisfying than the first two, be prepared to put some serious effort into it.
A hi-fi retailer who is expert at LP player set-up is an excellent option. Where do you live? Even if you're not in Southern California, Brian Berdan at Audio Elements in Pasadena is as good as they come. I would give him a call and see if he can help you. He sells excellent gear, and is a world-class set-up man.