Smooth Wall Collision Detection

Question

I've been working on a small game, and I've been stuck on the collision part. During the collision, when my player hits a block, he stops just as it should, but it then doesn't allow me to another direction (ex: the collision happens on the y axis, it won't allow me to move along the x axis). What I want is to be able to move along the wall while still moving into it (If you know what I mean). I've checked here and here, but they aren't what I'm looking for.

EDIT: The new code allows me to move along the axis's fine, but I'm getting some different problems (shown in pictures)

This is the NEW code for my collision detection:

if(key[KEY_A])
{
    if(tileMap[(x/zoomLevel)-1][y/zoomLevel] < 1 && tileMap[(x/zoomLevel)-1][(y/zoomLevel)+1] < 1)
    {
        if(tileMap[((x/zoomLevel)+1)][y/zoomLevel] < 1 && tileMap[((x/zoomLevel)+1)][(y/zoomLevel)+1] < 1)
        {
            x-=moveSpeed;
            dirX = 0;
        }
    }
}
else if(key[KEY_D])
{
    if(tileMap[(x/zoomLevel)+1][y/zoomLevel] < 1 && tileMap[(x/zoomLevel)+1][(y/zoomLevel)+1] < 1)
    {
        x+=moveSpeed;
        dirX = 1;
    }
}

if(jumping == false)
{
    if(tileMap[(x/zoomLevel)][(y/zoomLevel)+2] < 1)
    {
        y+=gravSpeed;
        gravSpeed*1.2;
        dirY = 1;
    }
}
else
{
    if(tileMap[(x/zoomLevel)][(y/zoomLevel)-1] < 1)
    {
        gravSpeed = 4;
        y-=gravSpeed;
        dirY = 0;
    }
}

Thanks to anyone who can help me with my problem!

And here are some picture based on the new collision:

Falling on ground:

Moving to the left when on top of a block:

Moving to the left:

Jumping when moving right:

Answer 1

I've been implementing some very similar collide-and-slide collision detection and resolution. http://metareal.net/blog/wp-content/uploads/2015/08/2015-08-21.jpg

The main key I found was to handle X and Y separately. So, your code might be modified something like this (a little freeform but you should get the idea):

// step 1, modify X
prevX = x;
x += movementX;
work out top, left, bottom, right
if(new span collides with anything) x = prevX;

// step 2, modify Y
prevY = y;
y += movementY;
work out top, left, bottom, right
if(new span collides with anything) y = prevY;

In certain cases, the order of checking x and y will make a difference. Specifically, heading directly onto a colliding corner will now favor X movement. This is probably ok, but can be changed if you need. (Literally a corner case!)

Hope that helps a little.