Your plant looks like a Euphorbia (leuconeura I think?), which is technically not a Madagascar palm (Pachypodium lamerei) but is also from Madagascar. These plants tend to get a little leggy, especially indoors, and that may contribute to the instability you are observing, but they are kind of top heavy to start with.
You can put the plant into a bigger pot if you like, but try to find one that is wider than deep, otherwise underfill whatever you do have. A nice heavy ceramic pot ought to provide a good anchor for the bushy top growth. Do not overpot the plant; it's better to go up stepwise in small increments rather than make big leaps all at once. Make sure the new pot has a hole at the bottom. Use regular potting soil mixed with an equal volume of perlite/pumice/equivalent to make sure the soil has excellent drainage.
I would not bury the skinny part at the base of the stem. That is the hypocotyl (seed stem) of the plant and you may run the risk of rotting the plant if you bury it. I think a nice heavy pot will better serve your needs anyway.
The plant looks quite healthy, but be aware that it enjoys a lot of light indoors, basically as much light as you can provide. More light will help encourage a compact form in the years going forward and work to your favor in avoiding rot. Using soil with good drainage will also help you avoid rot. Water well when the soil has gone dry or almost dry at depth, but there is no advantage to allowing the soil to sit bone dry for any extended period of time.
In the winter it will probably have far fewer leaves and will probably enjoy less frequent water than now, which is basically prime growing season.
