I live around there and worried about the same thing, but when I went to the open days I felt a lot more positive about it.
A lot of the kids around the area you mention go to St Barts (where ours goes). As it's a CofE school it doesn't register on the catchments map but in practice it has a reasonable catchment area. There are loads of kids on many of the roads off Venner Road down to Newlands Park who go there for instance.
Kelvin Grove probably too far catchment-wise, but worth a visit; seemed nice.
Adamsrill does, as you mention, have a large catchment area. I liked it when I went round at open day; friends have kids that go there who are very happy with it. Bit of a trek, obviously, but then so would Alexandra be in the first few years.
Harris Crystal Palace could well be within your catchment area this year; I liked it on an open-day visit. They struck me as a bit warmer and friendlier and I got a nice vibe from the place. Harris Kent House also might just about be within yr catchment area; I thought it was impressive at open day but felt a bit strict and results-focused.
It's not impossible you might get a place at Alexandra, so if that really is your favourite there's no harm in sticking it down as first choice.
Only local school I really didn't like much was St John's on Maple Road.
But don't take my word for it: you really do need to go to the open days and trust your instincts.
Don't forget that catchment areas vary a lot year to year: there might just have been a lot of siblings at a given school in any one year (who are prioritised a place). Don't forget that schools offer places without any knowledge of priority, so that means you can put your favourite as first choice however unlikely it is you will be offered it: you will be still be safe in the knowledge that a nearby-and-more-likely school will still offer you a place if they have one even if it's your 3rd, 4th or 5th choice. Make sure you use all your choices.