Multiplet shape in proton NMR of morpholines
It looks as if the NMR of morpholine is an AA′XX′ spectrum (the chemical shift difference is 0.80 ppm, or 320 Hz on your spectrometer, two orders of magnitude larger than the coupling constant).
Unlike linear AA′XX′ systems where the bonds can rotate, in the morpholine it very much has a fixed conformation.



In this fixed conformation, the protons are predominantly in a 'gauche' type arrangement due to the shape of the chair, this is where the appearance of the 'triplet' comes from in the NMR spectrum.
The image below is from Hans Reich's NMR notes, and shows some simulated NMR spectra depending on the gauche/anti ratio (Karpus equation), alone with a true spectrum of cyanomorpholine (which you can see looks basically like your spectrum above).


Source: Hans Reich (wisc.edu)


As a complete side-note. In TopSpin (which is what it looks like you used to process your NMR data) you can change the window function to use a Gaussian function, for strong spectra (which it looks like you have), this can help get a bit more resolution (at the expense of S:N- peaks on the baseline will disappear) such that what looks like a broad slightly messy spectrum actually gives resolved multiplets where the J values can be calculated.
In TopSpin, this can be done by entering the wm command; select &quot;gaussian&quot; from the drop-down list, and enter the two required parameters LB and GB (a decent starting choice is -2 and 0.2). The modified FID will now appear; this can be Fourier transformed using ft and automatically phase-corrected using apk to give the modified spectrum. [Even shorter is to sequentially type the commands lb -2, gb 0.2, and gfp into the command line.]
In MestreNova, the apodisation interface may be brought up from the Processing menu (or directly with the hotkey W) and the relevant parameters set in the same fashion.