Say for example in the beginning I have a typical "singing" melody, later in the song I have a more instrumental guitaresque solo, and then at the highest tension point of the song I have a sort of a breakdown melody.

When I listen to other songs these different parts don't seem to be continuous (as in my case) but they carry an overall style of the band i.e. if it's blues all these different parts will sound bluesy etc.

My concern is that my phrasing for each of these parts is largely arbitrary and I did not think of trying to make them connect in a continuous manner (although they transition well), basically I can substitute the instrumental guitaresque solo with another and the song isn't any better or worse.

Of course the different parts do carry the style, however the "singing melody" isn't as sad sounding as the other two, so there is a bit of variation in the mood.

So my question is when constructing these different parts of a song is it enough to ensure that they maintain the mood and style of the song or is there something more to it?
There's no trick or formula for writing a song. Being able to write something comes from a lot of experience and even more failure.

If the parts of your song do not go together, that is generally considered bad, though you may be going for that kind of effect.

Let your ear guide you; if you think it's no good, then it probably isn't any good.
There's no reason why all the parts need to be related. However, unless you're trying to give the effect of a sudden change, it's often better to use something to link them continuously. On the other hand, if you want that sudden change (from my experience often used in death metal), go right ahead.

This is definitely something where you should judge it more on how it sounds than what convention dictates. And remember that there's often something in music theory that would prove your decision right, you just haven't come across it yet.
Sound advice

However is it better to write all melody/solos first and work on the other sections separately?

My thinking is that when you wright all melodies and solos at once you can better gauge their relation rather than writing the song in a linear manner.
If you think that your song is sounding too through-composed (just individual sections stitched together instead of one song), maybe repeat a section towards the end, or hint to earlier riffs. The last movement of Beethoven's 9th Symphony actually quotes all three of the previous movements to remind the audience what they've been listening to for the past two hours.