The purpose of the priority rules, I believe, is there because some nurseries/schools are not suitable for some adopted children, whereas some are. So rather than being faced with the choice of 'unsuitable nursery/school' vs. 'nothing/home education', adopters should be able to send their child to a suitable setting, even if there is only one single one around for miles and miles that is actually suitable, and even if that one is oversubscribed.
I don't think the LA is obligated to help you, but they should, really, especially if it is kind of 'their idea' in the first place. And I think they CAN help you, if they want - though I'm not sure. I do know that they can MAKE a school offer your child a place, even if the school is full and doesn't want to. But I believe the situation regarding nursery school places is murkier. I think you need to motivate your SW (or maybe littledot's SW) to research what powers she actually has, let her find out how she can help you with this.
However FWIW DS' nursery school (which is a privately run one) said that they tend to have free spaces in September, less so in January, and hardly any in April. Reason being, the free entitlement (the general one, not the LAC one) kicks in the TERM after 3rd birthday, which is fairly evenly divided across the year - so new kids arrive all through the year. But they ALL leave at the end of the summer term, to start big school. So a whole year worth of children leaves end of July, but only a term worth of new children arrives in September.
So I guess what I'm saying is, have you tried some small privately run nursery schools? Rather than the LA run ones, or those associated with a primary school.
I'd say if possible (I know you're fairly busy and exhausted these days) - try to get this sorted soon (and tell the SWs that too!). Term ends in a couple of weeks or so, it will be hard to get hold of the relevant people afterwards, and of course no more chance to actually go and view the places.
(That said, it would probably give you peace of mind to have things sorted, BUT you could of course just wait and get it sorted once term starts up again in September - things aren't likely to change much (in terms of available places) between now and then, and if you do get LAC preference, you will still get that preference in September!)
I hope you're resting up; tomorrow is a new day, I hope it will be better!