The Conservative turn is actually a bit odd. It'd be like if the Democrats had gotten taken over by the Dixiecrats and everyone else got booted out of the party.
Certainly it can continue as a Moderate/Liberal controlled party, but whether or not you can get them to support the things you list is up in the air. Frankly, as I've said before, those are part of the political muck of distinction among factions. None of those are necessarily Liberal or Conservative, although they are features shared by the average member of either faction respectfully. You can oppose gay rights and abortion, and be a New Deal Liberal extraordinaire and a supporter of the Welfare state and whatever. They've only become part of a litmus test really from generalization.
The parties, first off, were also big tents with factions. Whatever faction or alliance of factions was in control was that which lead the party plank. In the Democrats, you had the Northern Liberals, the Conservative Dixiecrats, and the Southern Democrats who were New Dealers and not foaming at the mouth about segregation (whether they were against it or just pragmatic segregationists). In the GOP, you had the Liberals/Moderate alliance, and the Conservatives. In both parties, the Liberals/Moderates were the mainstream and dominant, they just approached issues from different vantage points between the parties. It is important to point out that the political consensus post-WW2 up to the Reagan era was solidly Liberal. The Dixiecrats were marginalized, often pain in the butts, and were only placated through some concessions and their own hatred of the Republicans. The Conservatives in the GOP were the Goldwater and later Reagan lead faction; they were limited and in the west. As factions were generally common in the past, the current state of polarization between the parties now is a bit odd and testament to the increasing weakening the parties have undergone in the past few decades.
If you want to avoid a Conservative takeover, it'd be good to stave off the fallout of the late 1960's. Basically the whole evolution of modern politics evolved from Vietnam, radicalization, disillusion, and so forth that snowballed on from there until everybody broke down and were willing to open the door to Nixon and Reagan, with all the muck inbetween complicating things up further.