'Pushing' is wrong here. However, you could possibly say 'I'm really pushed this weekend' and it would be understood, at least in British English. The phrase 'pushed for time' exists, meaning as you say, that you have a tight schedule and a lot to do in a short time. If you said 'pushed', people would probably know that you meant 'pushed for time'.

In the sentence you are using, pushing should be replaced with "Busy". So maybe say, I am really busy this weekend because I have lots of assignments to complete. Or I will be very busy this weekend because i will have a lot of assignments to complete. 1. Pushing doesn't mean tight schedule as a general rule. Pushing means: to press upon or against (a thing) with force in order to move it away. 2. to move (something) in a specified way by exerting force; shove; drive: to push something aside; to push the door open. 3. to effect or accomplish by thrusting obstacles aside: to push one's way through the crowd. 4. to cause to extend or project; thrust. 5. to press or urge to some action or course: His mother pushed him to get a job. http://dictionary.reference.com/browse/pushing?s=t