Per rules as written, no. The spell clearly says either one or the other, not both.

However, if a DM was to allow it, it probably wouldn't break much. The biggest concern would be that stealth against the target would likely be an automatic success, but that doesn't mean much when the target clearly knows that its senses are being suppressed.

Maybe...

As Artur Biesiadowski mentioned in a comment, it is questionable whether an elemental could be considered an object or surface for the Meld Into Stone spell to work. Elementals are usually referred to as creatures, but some are not considered very much different from the element they're made of. Consider this section from the Monster Manual (page 7, emphasis mine):

Elementals are creatures native to the elemental planes. Some creatures of this type are little more than animate masses of their respective elements, including the creatures simply called elementals. Others have biological forms infused with elemental energy.

The first sentence clearly identifies all elementals as creatures, but the second sentence makes it sound like some of them blur the line between object and creature. If my personal interpretation of the rules is correct, by RAW it would not work. However, your DM might disagree, or would like to make an exception to support your creativity.

...but even then it has its issues.

If allowed by the DM, the sequence of effects you described would result with you being encased in an elemental under your control. It might not be able to exactly "conform to your movements", seeing as you'll be standing inside it, but it'll follow your regular commands well enough.

The timing of Planar Binding would have to be perfect, though, to fit the 1 hour casting time in the 1 hour duration of the Conjure Elemental spell, but it could be made to work. The cost in gold is nothing to scoff at, but cast it at a higher level and it'll last you a while. Still, there are certain issues to consider:

1. Issuing commands

The biggest issue with this plan lies in the nature of the Meld Into Stone spell. Specifically, this section of the spell description (PHB page 259, emphasis mine):

While merged with the stone, you can’t see what occurs outside it, and any Wisdom (Perception) checks you make to hear sounds outside it are made with disadvantage.

This means you can command the elemental to do whatever you want, but you're virtually blind to your surroundings and are thus unable to command it effectively.

You can overcome this issue by using another spell, item or familiar to give you vision from outside the elemental. The other party members might be able to help, too.

2. Concentration

There is a small section of the Planar Binding spell that I find potentially troublesome (PHB page 265):

If the creature was summoned or created by another spell, that spell’s duration is extended to match the duration of this spell.

That makes it sound like the Conjure Elemental spell would still be in effect, and it's a spell that requires concentration. You would not be able to use any other spell that requires concentration, and would still be vulnerable to any effect that may break your concentration from within the stone. A spell like Shatter could still damage you from within the stone (and is also effective against your elemental).

It is quite likely that by RAI you shouldn't have to maintain concentration after casting Planar Binding, but that's something for your DM to decide.

Additionally, you might require an inverted Magic Circle to contain your elemental during the casting of Planar Binding. The SRD mentions that casting any spell with a casting time longer than 1 action requires Concentration (SRD page 101, emphasis mine):

When you cast a spell with a casting time longer than a single action or reaction, you must spend your action each turn casting the spell, and you must maintain your concentration while you do so. ...If your concentration is broken, the spell fails, but you don’t expend a spell slot.

This would mean that your elemental would break loose the moment you start casting Planar Binding, and would turn hostile (PHB, description of Conjure Elemental, page 225):

If your concentration is broken, the elemental doesn’t disappear. Instead, you lose control of the elemental, it becomes hostile toward you and your companions, and it might attack. An uncontrolled elemental can’t be dismissed by you, and it disappears 1 hour after you summoned it.

To my knowlege, such a rule is not contained in the PHB. But if it is accepted, it would solve the initial conundrum with concentration. If you can't concentrate on Conjure Elemental and cast Planar Binding at the same time, then all you're doing is binding an angry elemental that you trapped in a Magic Circle, and thus are free to concentrate on anything after the binding is complete. The elemental might still remain hostile, however, which would activate this section of Planar Binding:

The creature obeys the letter of your instructions, but if the creature is hostile to you, it strives to twist your words to achieve its own objectives.

3. Health management

You would have to find a way to reliably recover the elemental's HP, and be careful to try to step out of the elemental before he would be dealt lethal damage. As per the Meld Into Stone spell, if the elemental is destroyed while you're still in it, you would be ejected out, suffering 50 bludgeoning damage and knocking you prone.