Did you always had this issue or is it new ?
I don’t think your host is on target here, it’s not a memory or CPU issue MODx is efficient in this regard. Default memory limit is generally 8Mb and MODx runs fine with this.
I suspect it has something to do with cookies or sessions, since the login page uses them and I can’t see any other explanation for this.
Now about your host limitations : what are they ?
We need more info. Since you can’t login in the manager, you’ll have to create a phpinfo file manually :
1) With your favorite text editor, create a file and put this code in it :
<?php
phpinfo();
?>
2) Save it, name it info.php or whatever
3) Upload it somewhere safe (not at the root, preferably in a folder with a index.html file so that people can’t list the directory and see your config unless they have the adress you’ll give us).
4) Come back here an post the adress, so that we can check your server config.