You shouldn’t have a problem with Immigration. But, as you said, the airlines are supposed to deny boarding if you don’t have an onward ticket. Some airline clerks ask, some don’t. If they do ask, you can “rent” an onward ticket from a website or you can buy a refundable ticket, then cancel it after your flight.
I said any Immigration office. They’re all over the country. Go to the nearest one. Some require you to be on your last week. Others will let you extend sooner.
You need to hang onto your boarding pass when you arrive by air. You don’t need a boarding pass when you enter by land. You are allowed 2 entries by land or sea in a calendar year.