Just get a re-entry permit before you leave Steph, that will keep your admitted until date valid for 90 days from 3rd Feb. Re-entry permit is 1500 baht
In the event of being stopped and then you explaining your situation you may actually be shooting yourself in the foot as you will be explaining that your not actually a tourist and admitting to having the incorrect visa. You're best off with the correct visa especially with family involved. You can get 1 guardian visa for each child
With what you have ie birth cert you can get a single entry 90 day visa. You can then extend that for 2 months inside Thailand to get a maximum of 5 months. If you want longer you have to go through a court process to legitimise paternity. If you do that you can extend for a year provided you meet the financial requirements. You can then also extend annually thereafter