Yes it would make it much easier. There may be issues with the child's father not allowing him to be taken out of Thailand (depending on any custody agreements). But adoption is the best route.
The best advice is if you want to stay in Thailand long term, get a proper visa. If you do not have the finances or qualifications for a long term visa, then Thailand is not the answer for you. Sooner of later you will run out of options and immigration will get suspicious.