Yes the boat will shift, but it does that every time the trailer tilts on the road anyhow. I wouldn't stand under it but I would expect it to stay put.
WRT jacking the trailer, was it professionally built? If so, see if the maker can tell you the jacking points they recommend. If not, look for strong points near the axle, perhaps between the tires, or under the axle/suspension.
What I would most strongly suggest is buying a pair of jack stands that are rated well above the combined weight of the loaded trailer. Once you have one wheel up, about an inch off the ground, insert a jack stand just outboard of that wheel. Now repeat with the other wheel, so there are a pair of jackstands taking all the weight.
Assuming you loosened the lug nuts just a little beforehand, you can now spin 'em off, one tire at a time, and replace them. The tires should probably be chocked--all four of them, both forward and rear--to make sure the trailer doesn't shift while jacking.
As to the jacking points...BMW finally stole my idea, get a 3" copper pipe cap, find the jacking points, epoxy
or 5200 the cap to each jacking point so now the jack pretty much has to be centered in the right place. Or, at least gob on some blaze orange paint
to make it real obvious for next time.
And of course, you'll want to store those jack stands somewhere on the trailer. If you don't have a box of some kind attached to it already.
This is also the perfect time to install a Grease Buddy on each bearing, or at least, re-lubricate and inspect them.
If jacking, chocking, all that stuff is news to you? Even if you don't have an auto club membership, it might pay to have someone in the tow business top by, at their convenience, to show you just how to do it. If you feel any uncertainty about it--there's nothing like seeing it done the right way first. $20 and a cold six, someone ought to be willing to stop by when things are slow.
And if you put just a dab of antiseize on each wheel stud before you put the lug nuts back on? Lasts over ten years, and you'll never have a frozen lug nut again. You should, btw, also be using a torque wrench to put on lug nuts.