N.K.
I also have a 2 year old son who is very attached to me. But he can play by himself for some time while I am cooking or doing something. I work full time and he goes to daycare, so I do not expect him to play by himself after I pick him up in the evening :-) because we miss each other during the day and he is usually very hungry at that time. At this age and with this attachment, I would not expect a child to play by themselves for a long time but you should be able to get short breaks... Here is what I learned by experience:
-Do not expect him to play by himself if he is hungry or tired (or sick etc).
-Cuddle and spend time together first before trying to leave him to play by himself
-Start playing together. Most times they are not good at starting to play even if they have toys right in front of them. Say, you start building a tower together, or read a book, or start coloring etc...
-Tell him what you need to do and that you will be right back. Tell him what you expect him to do. ("Mommy needs to turn on the stove, and will be right back. You keep playing")
-You can try just stepping out the door and coming right back, to start with. Then gradually increase the time. He will need to learn to play by himself gradually.
-You can also involve him in what you are doing. He can help set up the table, do some cleaning etc. At this age, they really like to "help". Usually it is not a "help" but even increases the time needed to do something :-) but still you can get the thing done. When my son sees what I am doing and gets involved somehow (like stirring the pot, putting plates on the table etc) he sometimes prefers to go inside and play by himself instead :-)) Also explain to him that what you do needs to be get done and he can help if he wants (Such as mommy is cooking x, so we can eat it for dinner).
-Always praise him when he plays by himself, even not crying when you step out.
About the dad issue; we also had a similar situation, which got better now that his dad is playing with him more. If he plays with him more and has fun with him, he would prefer him more. As simple as that. Your husband may be feeling kind of rejected as he prefers you all the time but he needs to get over that feeling and get bonded with your son more. In a matter-of-fact way, without getting emotional, you can talk with your husband and simply say that your son needs more dad-son time and he will prefer you more if he spends more time with him. He is just a child and prefers to play/have fun with whomever he can. Your husband will need to be involved more when the second baby comes otherwise you will go crazy! There are games dads are better than moms, like playing ball, wresting etc :-) so they can have some fun while you do some chores.
Hope things will get better for all of you soon! I should say I still carry my son around a lot as he loves that, and it also doubles as weight lifting exercise :-))