Madame all M&S bra measurers will be first up against the wall, followed by their bra manufacturers.
Debenhams or John Lewis are much better, or independent shops quite often.
Yes, a proper fitting bra will feel tighter than what you're used to around the body, because the band should provide most of the support, not the straps. Like jeans feel tight and leave a mark on your tummy (or is that just me ? 
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When you measure around the body under your boobs, you should be wearing that band size, but it's all just a starting point. If you're well padded like me, you might size down a bit, if not you might size up slightly. The key is to try a bra on backwards without the straps over your shoulders. How far away can you pull the band ? Should be a couple of inches at most. Any more than that is too big. If the same bra then feels too small when you wear it right way round, it's because the cups are too small and are stealing fabric from the band.
When I first got fitted properly, it took ages to get the right size and style, because a lot of bras just don't work for me. Anything with a wire that comes right up in the centre (which is a lot of styles) digs right in, and I need a very wide wire to fit all my side-boob in. But other women have narrower boobs, so wide wires wouldn't work. So if you measure properly and try that size and it feels all wrong, it may be the wrong style, not the wrong size. I wear Bravissimo Satine, it's literally the only comfy style I've found.
Ahem bra rant over