If you can prove that you have been living together for one continuous year yes you can be included in her pnp/express entry. You would need proof like being included in your lease, filing taxes as common law which you should have done for a few years if you have lived together for 3 years, utilities in either/both your names, mail stating both your names, address, dates starting from the beginning of the one year that you lived together, etc. You should not drop your asylum claim because there are no guarantees until you actually have PR. It could take a long time to get PR and unfortunately relationships break up.