We will first analyze the explanations offered by the Rishonim for the practice of waiting six hours (or however long one’s minhag requires) between eating meat and milk and see which apply to your case. We will then refer to some of the Acharonim’s rulings to confirm that the analysis is applicable in practice.
The gemara[1] requires one to wait the amount of time in between typical meals between eating fleishig food and subsequently eating milchig food, but not vice versa. Rashi[2] explains that a certain fattiness comes out of the meat that one eats, and this remains in the mouth for quite some time. The Rambam[3] explains that the issue is the possibility that some meat will remain between the teeth.
Many authorities[4] discuss practical differences between these explanations. One difference occurs if one chewed meat but removed it before swallowing. The Rambam would require waiting because it is possible that some meat became stuck between the teeth. Rashi, however, would not require waiting because the taste lingers only when one swallows the meat.[5] Another difference pertains if one finds meat between his teeth after six hours. The Rambam apparently assumes that by then the meat is no longer problematic. According to Rashi, however, it is assumed to pose a problem. We accept the stringencies that follow from both explanations.[6]
If we posit, as above, that chewing does not pose a problem according to Rashi, then briefly having a fleishig crumb on one’s tongue certainly does not. Similarly, according to the Rambam, it is not logical to be concerned about meat between the teeth if the taster did no chewing at all.
Our case is quite similar to one discussed by early authorities in which after eating and waiting six hours, someone found and removed a piece of meat from between his teeth. In both cases, meat was in the mouth recently and was then removed, and in both cases, there is no concern for the impact of a recent eating. The Shach[7] says that even according to Rashi’s opinion that meat is still considered fleishig after six hours in the mouth, one does not have to wait another six hours from that point. Nevertheless, the Rama[8] maintains that in this situation, one should rinse his mouth before eating milchig. The Aruch HaShulchan[9] explains the Rama’s rationale: it is not reasonable that a mouth that had meat in it one moment could have milk in it the next moment without taking steps to remove the residue.
Most poskim[10] assume that the required manner of removing the residue is the same as when one wants to eat fleishig soon after eating milchig. In that context, we pasken that one needs to rinse the mouth with a liquid (hadacha) and eat a solid food to absorb the remaining taste (kinuach).[11] According to our analysis, then, you could have continued your milchig meal after rinsing your mouth and eating a pareve food. You should also have made sure there was no fleishig residue on your hands by either checking them or washing them.[12]
Despite this conclusion, general practice is sometimes stricter than analysis would indicate, as we will show with an example that resembles your case. The Pri Megadim[13] demonstrates that if one only chewed an otherwise pareve food that had absorbed some meat gravy, he has no inherent reason to wait six hours. Nonetheless, the Pri Megadim claims that due to the concept of lo p’lug (not distinguishing between similar cases that fall into the same category), one should wait six hours in this case. The poskim[14] accept this stringency, which raises the possibility that in our case, as well, we should follow a more stringent practice than our analysis would indicate. Nevertheless, the poskim rule regarding our case that one who merely tasted food with his tongue is not treated like one who chewed it, and he does not need to wait six hours.[15] However, both kinuach and hadacha are required.[16]
(Your situation includes additional mitigating circumstances that make leniency easier. However, even without them, one does not have to wait before eating milchig, and if necessary, it might even suffice to wash one’s mouth or eat something in between.[17])