Primary suspect rear shock. I assume it was you asking me about the spring?
Which spring did they end up putting in and how did they set the preload?
I'd recommend to get the rear shock setup sorted first before checking and modifying anything else.
Re. the front end: The OEM springs are quite lekker progressive (precisely spoken twin rate) ones and work well for the average rider.
Do NOT use additional preload spacers on twin rate or progressive rate springs unless you really know what you are doing (means unless you've calculated the spring)! Not for the OEM ones and also not for aftermarket ones!
The twin rate spring is supposed to work primarily on the soft part when riding normally. The soft part of the spring goes coil bound just below your normal supension sag making sure your ride is comfortable and the nose does not dive excessively when you drop anchor.
If the springs are too soft and you increase the preload on the spring the soft part will go coil bound too early, you basically convert the spring into a linear rate one. Not what you've paid for

You can calculate the takeover of a twin rate spring where the soft part goes coil bound. Ideally your sag should be some 30% whilst yourt takeover of the spring should be at about 40% - 50% compression. Preloading the spring will not only reduce the sag (this is what you want) but also take the takeover further up (this is what you don't want assuming the spring is correctly sized for the bike).
If your front is too soft you have various options:
Linear rate springs: Cheapest. Relatively easy and quick to get. (you can often find and adapt suitable MX bike fork springs).
Disadvantage: You have to go relatively hard with the initial setup to keep the bike from bottoming out and diving excessively. Not an issue with an MX where lap times & best match of spring and damper count but somewhat less than ideal on a DS bike which you might wish to ride all day and over 100's of km's.
Progressive / twin rate aftermarket. I've been selling them for close to 20 years and still have to see a bike where they don't make an improvement.
Progressive / twin rate custom build in the unlikely case that the above does not work. IMHO this will only be the case if the rear is too stiffly sprung and you can't get the preload set correctly.
One more thing about preload spacers and fork springs:
Always calculate the block length of your spring & spacer assembly and make sure it's less than the space between the bottomed out damper piston and fork cap. If it isn't and you bottom out your forks you will convert them into scrap metal. You will most probably destroy your damper rod and damper piston in this bottom out, the forks will with all likelihood not rebound and about 0.3 seconds later you will be able to see your rectum from the inside

A few more notes:
It is not true that low suspension setting DR650SE's are tendencially wobbly. Not if you follow procedure and lower the bike according to factory spec / the workshop manual. Most workshops don't and that's where the problem starts

Check the steering head bearing on your new DR650SE. All my seven DR650's came fron the factory with grossly overtightened steering heads. This wioll cause poor handling and lack of steering precision, high speed instability and it will destroy the bearings within 10,000km - 15,000km. And yes they are expensive and expensive to replace.
Ride knobblies a bit harder than factory spec especially at high speeds and on hardpack and tar. I found 1.7 bar (front) and 2 bar (rear) to work very well on my DR's. Keeps the bike stable up to top speed and helps tires to run cooler and last longer. I don't run my DR's softer than 1.7 bar at all as I don't do endless and bottomless sand. Works well for me and I never dinged a rim. Use it - don't use it. If you're big (I think you are) you can go a tad higher with tire pressures to account for more wheel load (1.8 / 2.3bar).
If all the above is set and in spec. and the bike still wobbles check if the front wheel is fitted correctly (tighten axle first then tighten clamp), the rear wheel is fitted straight (both issues I would recommend to check on a new bike

) and that the wheel bearings are ok (no issue on a new bike).
Don't think "ride through it" is the solution to suspension problems.