, Stick probably offers the best solution, but I notice you say you do not have tongue-and-groove bits. Do you have a slot cutter? If not, take a leaf out of JOAT’s book, and make yourself a very simple dowelling jig. Instructions on the internet.
Provided you register the jig against the “bookmatch” faces of both boards (as opposed to face of one and back of the other), the faces should be well-aligned. Do a dry-fit before glue-up, and re-drill any dowel that is problematic. A slight longitudinal discrepancy will not matter, since you will be trimming to length anyway.
In my humble experience, the “affordable” dowelling jigs are not better than homemade.
Regarding the position of the glue-line,
1. Will it be visible inside the cabinet?
2. Will the board be weight-bearing?
3. Are there any other structural members to give support?
4. Exactly what material are you using? If by laminate you mean melamine-faced, it is darn difficult to get an invisible glue line. In that case, you might want to position the glue line as far towards the back as possible. Also, the factory edges will possibly be more chip-free than you or I can achieve by cutting. Ditto for the edging. If the laminate is wood veneer, it may not matter so much.
Perhaps post a sketch?