@negrogesic made a great post. This is more or less what I would suggest. Opiates like Morphine/Codeine are both readily soluble in water. You shouldn't need any complex processes to separate the desired Opiates from the plant matter. You're going to end up eating some plant matter though, that's unavoidable.
Once you have vigorously mixed the seeds and water, you will be left with your solution. As
@negrogesic was saying, you can do what you like with this solution. It can be consumed as is, distilled i.e. evaporated i.e. purged of moisture i.e. made more potent. If you leave the solution out, it will slowly evaporate on its own. This process can be expedited by using a fan to blow air on your solution, by putting it in the oven on a low temperature etc. Don't boil anything, as this can destroy your chemicals.
Your issue is that you're being made nauseous by all of the plant material. The best way to avoid this is by distilling your solution. Make it stronger so you have to consume less liquid, likely leading to less nausea. To speed evaporation, put the solution in a container that will give you a large, exposed surface area like a pyrex baking tray or what have you.
I'll say this, we know that commercial Opium is typically gonna have an alkaloid content of 10% Morphine/3% Codeine. This would give us ~80mg Morphine right? Now, if we're talking about Poppy Seeds, who knows? If I were to shoot from the hip here, I'd guess that less than 2% of the total weight of the seeds is going to be Morphine. So, let's say you gho through all of this, do the wash successfully and distill your solution, you would be getting 20mg Morphine at the absolute most.
For someone with not prior tolerance to Opioids, this would give them a pleasant buzz. For someone like yourself, it's not going to send you to Mars or anything.