PA28 STUDENT AND INSTRUCTOR TKOF INTO BLOWING SNOW IN ORDER TO GET HOME.

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TRAINING FLT HAD STOPPED AT D37 FOR A QUICK BREAK WHEN BLOWING SNOW WAS NOTED. IN THE INTEREST OF PREVENTING ANY ACCUMULATIONS ON THE BRAKES OR UNDERSIDE OF LIFTING SURFACES; WE WERE ROLLING WITHIN 4 MINS OF OUR LNDG. WX AT THE TIME OF LNDG WAS WITH A CEILING GREATER THAN 5000 FT MSL; VISIBILITIES OF 10+ NM TO THE W; AND UNLIMITED TO THE E. UPON OUR TKOF; IT WAS APPARENT THAT THE WX HAD REDUCED GREATLY. THE FORECAST WAS FOR BROKEN 020 WITH -SN. THE WX PREVIOUS HAD NOT INDICATED A GREAT REDUCTION IN WX AND WE HAD NO CONCERN BEFORE OUR TKOF OUT OF D37. THE WX HAD REDUCED TO APPROX BROKEN 007 WITH 3-5 SM VISIBILITY. WE LOST SIGHT OF D37 AND HAD TO SCUD RUN TO AVOID THE CLOUDS. AN IFR FLT PLAN WAS OPENED TO GET US BACK INTO GFK. WE RECEIVED VECTORS FOR THE ILS RWY 35L. DURING OUR VECTORS WE FLEW IN AND OUT OF BOTH CLOUDS AND SNOW; AND ICING WAS A WORRY AS PIREPS FROM HIGHER WERE RPTING ICING. WE LANDED IN GFK WITH NO INCIDENT OF ICING OR OTHER PROBS WITH THE ACFT. I UNDERSTAND THE SITUATION COULD HAVE BEEN MUCH WORSE AND SHOULD HAVE REQUESTED AN APCH BACK INTO THE ARPT WHICH I HAD LEFT; D37. THE CALL WAS MADE ON THE TIME OF GREATER ACFT SUPPORT AT GFK THAN D37 AS D37 IS UNATTENDED. HOWEVER; WITH THE FLT TIME OF .5 TO REACH GFK AND POSSIBLY AS LOW AT .3 TO GET BACK TO D37; IF ICING HAD OCCURRED; THE .2 COULD HAVE BEEN THE DIFFERENCE BTWN LNDG ON A RWY AND LNDG IN A FIELD. THERE WAS ALSO THE FACT THAT UPON LISTENING TO GFK APCH OTHER ACFT IN A SIMILAR CONFIGN WERE MAKING IT INTO GFK WITH NO ICING; CAUSING THE ILLUSION OF SAFETY. THIS FLT TURNED OUT TO BE LUCKY; BUT WITH OUTSIDE AIR TEMPS BELOW FREEZING; WHILE SNOWING; AND IN AN ACFT NOT EQUIPPED FOR ICING; IT COULD HAVE ENDED BADLY.