Both Uses

besiege
in
Love's Labour's Lost
(Auto-generated)

'So it is, besieged with sable-coloured melancholy, I did commend the black-oppressing humour to the most wholesome physic of thy health-giving air; and, as I am a gentleman, betook myself to walk.†
Scene 1.1 *

besieged = surrounded and cut-off supplies; or overwhelmed
Marry, thus much I have learnt; He rather means to lodge you in the field, Like one that comes here to besiege his court, Than seek a dispensation for his oath, To let you enter his unpeeled house.†
Scene 2.1

besiege = surround and cut-off supplies; or overwhelm

Definitions:

(1) (besiege)
to surround a place with armed forces and cut off supplies or attack; or more generally, to overwhelm someone with many requests, complaints, or attention, as if surrounding them and not letting up
(2) (meaning too rare to warrant focus)